{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2022-10-18T16:01:26+00:00"},"categoryId":33727,"data":{"title":"Pre-Calculus","slug":"pre-calculus","image":{"src":null,"width":0,"height":0},"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"parentCategory":{"categoryId":33720,"title":"Math","slug":"math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"}},"childCategories":[],"description":"Can learning (or reviewing) pre-calc really be fun? With our breakdowns and practice problems, it'll come pretty close.","relatedArticles":{"self":"https://dummies-api.dummies.com/v2/articles?category=33727&offset=0&size=5"},"hasArticle":true,"hasBook":true,"articleCount":206,"bookCount":3},"_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":206,"items":[{"headers":{"creationTime":"2016-03-26T15:11:52+00:00","modifiedTime":"2022-10-06T21:01:24+00:00","timestamp":"2022-10-07T00:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Pre-Calculus: Test the Roots By Long Division of Polynomials","strippedTitle":"pre-calculus: test the roots by long division of polynomials","slug":"how-to-guess-and-check-real-roots-2-testing-roots-by-dividing-polynomials-using-long-division","canonicalUrl":"","seo":{"metaDescription":"One way to test the real roots is to use long division of polynomials and hope that when you divide you get a remainder of 0.","noIndex":0,"noFollow":0},"content":"Once you have used the rational root theorem to list all the possible<i> </i>rational roots of any polynomial, the next step is to test the roots. One way is to use long division of polynomials and hope that when you divide you get a remainder of 0. Once you have a list of possible rational roots, you then pick one and assume that it’s a root.\r\n\r\nFor example, consider the equation <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48, which has the following possible rational roots:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370082.image0.png\" alt=\"image0.png\" width=\"453\" height=\"19\" />\r\n\r\nIf <i>x</i> = <i>c</i> is a root<i>,</i> then <i>x</i> – c is a factor. So if you pick <i>x</i> = 2 as your guess for the root, <i>x</i> – 2 should be a factor. You can use long division to test if <i>x</i> – 2 is actually a factor and, therefore, <i>x</i> = 2 is a root.\r\n\r\nDividing polynomials to get a specific answer isn’t something you do every day, but the idea of a function or expression that’s written as the quotient of two polynomials is important for pre-calculus. If you divide a polynomial by another and get a remainder of 0, the divisor is a factor, which in turn gives a root.\r\n<p class=\"TechnicalStuff\">In math lingo, the division algorithm states the following: If <i>f</i>(<i>x</i>) and <i>d</i>(<i>x</i>) are polynomials such that <i>d</i>(<i>x</i>) isn’t equal to 0, and the degree of <i>d</i>(<i>x</i>) isn’t larger than the degree of <i>f</i>(<i>x</i>), there are unique polynomials <i>q</i>(<i>x</i>) and <i>r</i>(<i>x</i>)<i> </i>such that</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370083.image1.png\" alt=\"image1.png\" width=\"159\" height=\"20\" />\r\n\r\nIn plain English, the dividend equals the divisor times the quotient plus the remainder. You can always check your results by remembering this information.\r\n<p class=\"Tip\">Remember the mnemonic device<i> </i><u>D</u>irty <u>M</u>onkeys <u>S</u>mell <u>B</u>ad when doing long division to check your roots. Make sure all terms in the polynomial are listed in descending order and that every degree is represented. In other words, if <i>x</i><sup>2</sup> is missing, put in a placeholder of 0<i>x</i><sup>2</sup> and then do the division. (This step is just to make the division process easier.)</p>\r\nTo divide two polynomials, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">Divide the leading term of the dividend by the leading term of the divisor. Write this quotient directly above the term you just divided into.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply the quotient term from Step 1 by the entire divisor. Write this polynomial under the dividend so that like terms are lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract the whole line you just wrote from the dividend.</p>\r\n<p class=\"child-para\">You can change all the signs and add if it makes you feel more comfortable. This way, you won’t forget signs.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down the next term.</p>\r\n<p class=\"child-para\">Do exactly what this says; bring down the next term in the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Repeat Steps 1–4 over and over until the remainder polynomial has a degree that’s less than the dividend’s.</p>\r\n</li>\r\n</ol>\r\n<p class=\"Tip\">The following list explains how to divide 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x </i>+ 48 by <i>x</i> – 2. Each step corresponds with the numbered step in the illustration in this figure.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370084.image2.jpg\" alt=\"The process of long division of polynomials.\" width=\"400\" height=\"394\" />\r\n<div class=\"imageCaption\">The process of long division of polynomials.</div>\r\n</div>\r\n(Note that using Descartes’s rule of signs, you find that this particular example may have positive roots, so it’s efficient to try a positive number here. If Descartes’s rule of signs had said that no positive roots existed, you wouldn’t test any positives!)\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">What do you have to multiply <i>x</i> in the divisor by to make it become 2<i>x</i><sup>4</sup> in the dividend? The quotient, 2<i>x</i><sup>3</sup>, goes above the 2<i>x</i><sup>4</sup> term.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply this quotient by the divisor and write it under the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract this line from the dividend: (2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup>) – (2<i>x</i><sup>4</sup> – 4<i>x</i><sup>3</sup>) = –5<i>x</i><sup>3</sup>. If you’ve done the job right, the subtraction of the first terms always produces 0.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\">Bring down the other terms of the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What do you have to multiply <i>x</i> by to make it –5<i>x</i><sup>3</sup>? Put the answer, –5<i>x</i><sup>2</sup>, above the –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>Multiply the –5<i>x</i><sup>2</sup> times the <i>x</i> – 2 to get –5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>. Write it under the remainder with the degrees lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You now have (–5<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup>) – (–5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>) = –31<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +88<i>x</i> takes its place.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What to multiply by to make <i>x</i> become –31<i>x</i><sup>2</sup>? The quotient –31<i>x</i> goes above –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The value –31<i>x</i> times (<i>x</i> – 2) is –31<i>x</i><sup>2</sup> + 62<i>x;</i> write it under the remainder.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b> You now have (–31<i>x</i><sup>2</sup> + 88<i>x</i>) – (–31<i>x</i><sup>2</sup> + 62<i>x</i>), which is 26<i>x.</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +48 comes down.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>The term 26<i>x</i> divided by <i>x</i> is 26. This answer goes on top.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The constant 26 multiplied by (<i>x</i> – 2) is 26<i>x</i> – 52.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You subtract (26<i>x</i> + 48) – (26<i>x</i> – 52) to get 100.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>top.</p>\r\n<p class=\"child-para\"><b></b>The remainder 100 has a degree that’s less than the divisor of <i>x</i> – 2.</p>\r\n</li>\r\n</ol>\r\nWow . . . now you know why they call it <i>long</i> division. You went through all that to find out that <i>x</i> – 2 isn’t a factor of the polynomial, which means that <i>x</i> = 2 isn’t a root.\r\n<p class=\"Remember\">If you divide by <i>c</i> and the remainder is 0, then the linear expression (<i>x</i> – <i>c</i>) is a factor and that <i>c</i> is a root. A remainder other than 0 implies that (<i>x</i> – <i>c</i>) isn’t a factor and that <i>c</i> isn’t a root.</p>","description":"Once you have used the rational root theorem to list all the possible<i> </i>rational roots of any polynomial, the next step is to test the roots. One way is to use long division of polynomials and hope that when you divide you get a remainder of 0. Once you have a list of possible rational roots, you then pick one and assume that it’s a root.\r\n\r\nFor example, consider the equation <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48, which has the following possible rational roots:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370082.image0.png\" alt=\"image0.png\" width=\"453\" height=\"19\" />\r\n\r\nIf <i>x</i> = <i>c</i> is a root<i>,</i> then <i>x</i> – c is a factor. So if you pick <i>x</i> = 2 as your guess for the root, <i>x</i> – 2 should be a factor. You can use long division to test if <i>x</i> – 2 is actually a factor and, therefore, <i>x</i> = 2 is a root.\r\n\r\nDividing polynomials to get a specific answer isn’t something you do every day, but the idea of a function or expression that’s written as the quotient of two polynomials is important for pre-calculus. If you divide a polynomial by another and get a remainder of 0, the divisor is a factor, which in turn gives a root.\r\n<p class=\"TechnicalStuff\">In math lingo, the division algorithm states the following: If <i>f</i>(<i>x</i>) and <i>d</i>(<i>x</i>) are polynomials such that <i>d</i>(<i>x</i>) isn’t equal to 0, and the degree of <i>d</i>(<i>x</i>) isn’t larger than the degree of <i>f</i>(<i>x</i>), there are unique polynomials <i>q</i>(<i>x</i>) and <i>r</i>(<i>x</i>)<i> </i>such that</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370083.image1.png\" alt=\"image1.png\" width=\"159\" height=\"20\" />\r\n\r\nIn plain English, the dividend equals the divisor times the quotient plus the remainder. You can always check your results by remembering this information.\r\n<p class=\"Tip\">Remember the mnemonic device<i> </i><u>D</u>irty <u>M</u>onkeys <u>S</u>mell <u>B</u>ad when doing long division to check your roots. Make sure all terms in the polynomial are listed in descending order and that every degree is represented. In other words, if <i>x</i><sup>2</sup> is missing, put in a placeholder of 0<i>x</i><sup>2</sup> and then do the division. (This step is just to make the division process easier.)</p>\r\nTo divide two polynomials, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">Divide the leading term of the dividend by the leading term of the divisor. Write this quotient directly above the term you just divided into.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply the quotient term from Step 1 by the entire divisor. Write this polynomial under the dividend so that like terms are lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract the whole line you just wrote from the dividend.</p>\r\n<p class=\"child-para\">You can change all the signs and add if it makes you feel more comfortable. This way, you won’t forget signs.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down the next term.</p>\r\n<p class=\"child-para\">Do exactly what this says; bring down the next term in the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Repeat Steps 1–4 over and over until the remainder polynomial has a degree that’s less than the dividend’s.</p>\r\n</li>\r\n</ol>\r\n<p class=\"Tip\">The following list explains how to divide 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x </i>+ 48 by <i>x</i> – 2. Each step corresponds with the numbered step in the illustration in this figure.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370084.image2.jpg\" alt=\"The process of long division of polynomials.\" width=\"400\" height=\"394\" />\r\n<div class=\"imageCaption\">The process of long division of polynomials.</div>\r\n</div>\r\n(Note that using Descartes’s rule of signs, you find that this particular example may have positive roots, so it’s efficient to try a positive number here. If Descartes’s rule of signs had said that no positive roots existed, you wouldn’t test any positives!)\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">What do you have to multiply <i>x</i> in the divisor by to make it become 2<i>x</i><sup>4</sup> in the dividend? The quotient, 2<i>x</i><sup>3</sup>, goes above the 2<i>x</i><sup>4</sup> term.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply this quotient by the divisor and write it under the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract this line from the dividend: (2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup>) – (2<i>x</i><sup>4</sup> – 4<i>x</i><sup>3</sup>) = –5<i>x</i><sup>3</sup>. If you’ve done the job right, the subtraction of the first terms always produces 0.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\">Bring down the other terms of the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What do you have to multiply <i>x</i> by to make it –5<i>x</i><sup>3</sup>? Put the answer, –5<i>x</i><sup>2</sup>, above the –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>Multiply the –5<i>x</i><sup>2</sup> times the <i>x</i> – 2 to get –5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>. Write it under the remainder with the degrees lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You now have (–5<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup>) – (–5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>) = –31<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +88<i>x</i> takes its place.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What to multiply by to make <i>x</i> become –31<i>x</i><sup>2</sup>? The quotient –31<i>x</i> goes above –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The value –31<i>x</i> times (<i>x</i> – 2) is –31<i>x</i><sup>2</sup> + 62<i>x;</i> write it under the remainder.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b> You now have (–31<i>x</i><sup>2</sup> + 88<i>x</i>) – (–31<i>x</i><sup>2</sup> + 62<i>x</i>), which is 26<i>x.</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +48 comes down.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>The term 26<i>x</i> divided by <i>x</i> is 26. This answer goes on top.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The constant 26 multiplied by (<i>x</i> – 2) is 26<i>x</i> – 52.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You subtract (26<i>x</i> + 48) – (26<i>x</i> – 52) to get 100.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>top.</p>\r\n<p class=\"child-para\"><b></b>The remainder 100 has a degree that’s less than the divisor of <i>x</i> – 2.</p>\r\n</li>\r\n</ol>\r\nWow . . . now you know why they call it <i>long</i> division. You went through all that to find out that <i>x</i> – 2 isn’t a factor of the polynomial, which means that <i>x</i> = 2 isn’t a root.\r\n<p class=\"Remember\">If you divide by <i>c</i> and the remainder is 0, then the linear expression (<i>x</i> – <i>c</i>) is a factor and that <i>c</i> is a root. A remainder other than 0 implies that (<i>x</i> – <i>c</i>) isn’t a factor and that <i>c</i> isn’t a root.</p>","blurb":"","authors":[{"authorId":9703,"name":"Yang Kuang","slug":"yang-kuang","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9703"}},{"authorId":9704,"name":"Elleyne Kase","slug":"elleyne-kase","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9704"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-633f6c3f1e5c1\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-633f6c3f1f1f6\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-10-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":167920},{"headers":{"creationTime":"2016-03-26T15:10:37+00:00","modifiedTime":"2022-10-06T15:09:45+00:00","timestamp":"2022-10-06T18:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Pre-Calculus: Finding the General Formula for the nth Term","strippedTitle":"pre-calculus: finding the general formula for the nth term","slug":"how-to-find-the-general-formula-for-the-nth-term-of-an-arithmetic-sequence-using-any-two-terms","canonicalUrl":"","seo":{"metaDescription":"Learn how to find the general formula for the nth term of an arithmetic sequence without knowing the first term or common difference.","noIndex":0,"noFollow":0},"content":"At some point, your pre-calculus teacher will ask you to find the general formula for the <i>n</i>th term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will be given two terms (not necessarily consecutive), and you will use this information to find <i>a</i><sub>1</sub> and <i>d.</i> The steps are: Find the common difference <i>d</i>, write the specific formula for the given sequence, and then find the term you're looking for.\r\n\r\nFor instance, to find the general formula of an arithmetic sequence where <i>a</i><sub>4</sub> = –23 and <i>a</i><sub>22</sub> = 40, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Find the common difference.</p>\r\n<p class=\"child-para\">You have to be creative in finding the common difference for these types of problems.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Use the formula <i>a</i><i><sub>n</sub></i> = <i>a</i><sub>1</sub> + (<i>n</i> – 1)<i>d</i> to set up two equations that use the given information.</p>\r\n<p class=\"child-para\">For the first equation, you know that when <i>n</i> = 4, <i>a</i><i><sub>n</sub></i> = –23:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + (4 – 1)<i>d</i></p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3<i>d</i></p>\r\n<p class=\"child-para\">For the second equation, you know that when <i>n</i> = 22, <i>a</i><i><sub>n</sub></i> = 40:</p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + (22 – 1)<i>d</i></p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + 21<i>d</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Set up a system of equations and solve for <i>d.</i></p>\r\n<p class=\"child-para\">The system looks like this:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370612.image0.png\" alt=\"image0.png\" width=\"199\" height=\"56\" />\r\n<p class=\"child-para\">You can use elimination or substitution to solve the system. Elimination works nicely because you can multiply either equation by –1 and add the two together to get 63 = 18<i>d. </i>Therefore, <i>d</i> = 3.5.</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Write the formula for the specific sequence.</p>\r\n<p class=\"child-para\">This step involves a little work.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Plug <i>d</i> into one of the equations to solve for <i>a</i><sub>1</sub><i>.</i></p>\r\n<p class=\"child-para\">You can plug 3.5 back into either equation:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3(3.5), or <i>a</i><sub>1</sub> = –33.5.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Use <i>a</i><sub>1</sub> and <i>d</i> to find the general formula for <i>a</i><i><sub>n</sub></i><i>.</i></p>\r\n<p class=\"child-para\">This step becomes a simple three-step simplification:</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= –33.5 + (<i>n </i>– 1)3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i> = –33.5 + 3.5<i>n</i> – 3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= 3.5<i>n</i> – 37</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the term you were looking for.</p>\r\n<p class=\"child-para\">In this example, you weren't asked to find any specific term (always read the directions!), but if you were, you could plug that number in for <i>n</i> and then find the term you were looking for.</p>\r\n</li>\r\n</ol>","description":"At some point, your pre-calculus teacher will ask you to find the general formula for the <i>n</i>th term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will be given two terms (not necessarily consecutive), and you will use this information to find <i>a</i><sub>1</sub> and <i>d.</i> The steps are: Find the common difference <i>d</i>, write the specific formula for the given sequence, and then find the term you're looking for.\r\n\r\nFor instance, to find the general formula of an arithmetic sequence where <i>a</i><sub>4</sub> = –23 and <i>a</i><sub>22</sub> = 40, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Find the common difference.</p>\r\n<p class=\"child-para\">You have to be creative in finding the common difference for these types of problems.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Use the formula <i>a</i><i><sub>n</sub></i> = <i>a</i><sub>1</sub> + (<i>n</i> – 1)<i>d</i> to set up two equations that use the given information.</p>\r\n<p class=\"child-para\">For the first equation, you know that when <i>n</i> = 4, <i>a</i><i><sub>n</sub></i> = –23:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + (4 – 1)<i>d</i></p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3<i>d</i></p>\r\n<p class=\"child-para\">For the second equation, you know that when <i>n</i> = 22, <i>a</i><i><sub>n</sub></i> = 40:</p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + (22 – 1)<i>d</i></p>\r\n<p class=\"child-para\">40 = <i>a</i><sub>1</sub> + 21<i>d</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Set up a system of equations and solve for <i>d.</i></p>\r\n<p class=\"child-para\">The system looks like this:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370612.image0.png\" alt=\"image0.png\" width=\"199\" height=\"56\" />\r\n<p class=\"child-para\">You can use elimination or substitution to solve the system. Elimination works nicely because you can multiply either equation by –1 and add the two together to get 63 = 18<i>d. </i>Therefore, <i>d</i> = 3.5.</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Write the formula for the specific sequence.</p>\r\n<p class=\"child-para\">This step involves a little work.</p>\r\n\r\n<ol class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">a.Plug <i>d</i> into one of the equations to solve for <i>a</i><sub>1</sub><i>.</i></p>\r\n<p class=\"child-para\">You can plug 3.5 back into either equation:</p>\r\n<p class=\"child-para\">–23 = <i>a</i><sub>1</sub> + 3(3.5), or <i>a</i><sub>1</sub> = –33.5.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">b.Use <i>a</i><sub>1</sub> and <i>d</i> to find the general formula for <i>a</i><i><sub>n</sub></i><i>.</i></p>\r\n<p class=\"child-para\">This step becomes a simple three-step simplification:</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= –33.5 + (<i>n </i>– 1)3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i> = –33.5 + 3.5<i>n</i> – 3.5</p>\r\n<p class=\"child-para\"><i>a</i><i><sub>n</sub></i><i> </i>= 3.5<i>n</i> – 37</p>\r\n</li>\r\n</ol>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the term you were looking for.</p>\r\n<p class=\"child-para\">In this example, you weren't asked to find any specific term (always read the directions!), but if you were, you could plug that number in for <i>n</i> and then find the term you were looking for.</p>\r\n</li>\r\n</ol>","blurb":"","authors":[{"authorId":9703,"name":"Yang Kuang","slug":"yang-kuang","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9703"}},{"authorId":9704,"name":"Elleyne Kase","slug":"elleyne-kase","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9704"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-633f17decd724\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-633f17dece54d\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-10-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":167814},{"headers":{"creationTime":"2016-03-26T15:06:54+00:00","modifiedTime":"2022-09-22T16:00:35+00:00","timestamp":"2022-09-22T18:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Finding the Cosine of Sums and Differences of Angles","strippedTitle":"finding the cosine of sums and differences of angles","slug":"applying-the-sum-and-difference-formulas-for-cosines-to-find-the-cosine-of-the-sum-or-difference-of-two-angles","canonicalUrl":"","seo":{"metaDescription":"In trigonometry, you can use the cosine sum and difference formulas to calculate the cosine of the sums and differences of angles.","noIndex":0,"noFollow":0},"content":"You can use the sum and difference formulas for cosine to calculate the cosine of the sums and differences of angles similarly to the way you can use the sum and difference formulas for sine, because the formulas look very similar to each other. When working with sines and cosines of sums and differences of angles, you're simply plugging in given values for the variables (angles). Just make sure you use the correct formula based on the information you're given in the question.\r\n\r\nHere are the sum and difference formulas for cosines:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372124.image0.png\" alt=\"image0.png\" width=\"249\" height=\"53\" />\r\n\r\nThe sum and difference formulas for cosine (and sine) can do more than calculate a trig value for an angle not marked on the unit circle (at least for angles that are multiples of 15 degrees). They can also be used to find the cosine (and sine) of the sum or difference of two angles based on information given about the two angles. For such problems, you'll be given two angles (call them A and B), the sine or cosine of A and B, and the quadrant(s) in which the two angles are located.\r\n\r\nUse the following steps to find the exact value of cos(A + B), given that cos A = –3/5, with A in quadrant II of the coordinate plane, and sin B = –7/25, with B in quadrant III:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Choose the appropriate formula and substitute the information you know to determine the missing information.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372125.image1.png\" alt=\"image1.png\" width=\"261\" height=\"27\" />\r\n<p class=\"child-para\">then substitutions result in this equation:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372126.image2.png\" alt=\"image2.png\" width=\"248\" height=\"40\" />\r\n<p class=\"child-para\">To proceed any further, you need to find cos B and sin A.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Draw pictures representing right triangles in the quadrant(s).</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 238px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372127.image3.jpg\" alt=\"Drawing pictures helps you visualize the missing pieces of info.\" width=\"238\" height=\"400\" />\r\n<div class=\"imageCaption\">Drawing pictures helps you visualize the missing pieces of info.</div>\r\n</div>\r\n<p class=\"child-para\">You need to draw one triangle for angle A in quadrant II and one for angle B in quadrant III. Using the definition of sine as <i>opp</i>/<i>hyp</i> and cosine as <i>adj</i>/<i>hyp,</i> this figure shows these triangles. Notice that the value of a leg is missing in each triangle.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">To find the missing values, use the Pythagorean theorem.</p>\r\n<p class=\"child-para\">The length of the missing leg in Figure a is 4, and the length of the missing leg in Figure b is –24.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Determine the missing trig ratios to use in the sum or difference formula.</p>\r\n<p class=\"child-para\">You use the definition of cosine to find that cos B = –24/25 and the definition of sine to find that sin A = 4/5.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Substitute the missing trig ratios into the sum or difference formula and simplify.</p>\r\n<p class=\"child-para\">You now have this equation:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372128.image4.png\" alt=\"image4.png\" width=\"251\" height=\"40\" />\r\n<p class=\"child-para\">Follow the order of operations to get this answer:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372129.image5.png\" alt=\"image5.png\" width=\"295\" height=\"45\" />\r\n<p class=\"child-para\">This equation simplifies to cos(A + B) = 4/5.</p>\r\n</li>\r\n</ol>","description":"You can use the sum and difference formulas for cosine to calculate the cosine of the sums and differences of angles similarly to the way you can use the sum and difference formulas for sine, because the formulas look very similar to each other. When working with sines and cosines of sums and differences of angles, you're simply plugging in given values for the variables (angles). Just make sure you use the correct formula based on the information you're given in the question.\r\n\r\nHere are the sum and difference formulas for cosines:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372124.image0.png\" alt=\"image0.png\" width=\"249\" height=\"53\" />\r\n\r\nThe sum and difference formulas for cosine (and sine) can do more than calculate a trig value for an angle not marked on the unit circle (at least for angles that are multiples of 15 degrees). They can also be used to find the cosine (and sine) of the sum or difference of two angles based on information given about the two angles. For such problems, you'll be given two angles (call them A and B), the sine or cosine of A and B, and the quadrant(s) in which the two angles are located.\r\n\r\nUse the following steps to find the exact value of cos(A + B), given that cos A = –3/5, with A in quadrant II of the coordinate plane, and sin B = –7/25, with B in quadrant III:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Choose the appropriate formula and substitute the information you know to determine the missing information.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372125.image1.png\" alt=\"image1.png\" width=\"261\" height=\"27\" />\r\n<p class=\"child-para\">then substitutions result in this equation:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372126.image2.png\" alt=\"image2.png\" width=\"248\" height=\"40\" />\r\n<p class=\"child-para\">To proceed any further, you need to find cos B and sin A.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Draw pictures representing right triangles in the quadrant(s).</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 238px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372127.image3.jpg\" alt=\"Drawing pictures helps you visualize the missing pieces of info.\" width=\"238\" height=\"400\" />\r\n<div class=\"imageCaption\">Drawing pictures helps you visualize the missing pieces of info.</div>\r\n</div>\r\n<p class=\"child-para\">You need to draw one triangle for angle A in quadrant II and one for angle B in quadrant III. Using the definition of sine as <i>opp</i>/<i>hyp</i> and cosine as <i>adj</i>/<i>hyp,</i> this figure shows these triangles. Notice that the value of a leg is missing in each triangle.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">To find the missing values, use the Pythagorean theorem.</p>\r\n<p class=\"child-para\">The length of the missing leg in Figure a is 4, and the length of the missing leg in Figure b is –24.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Determine the missing trig ratios to use in the sum or difference formula.</p>\r\n<p class=\"child-para\">You use the definition of cosine to find that cos B = –24/25 and the definition of sine to find that sin A = 4/5.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Substitute the missing trig ratios into the sum or difference formula and simplify.</p>\r\n<p class=\"child-para\">You now have this equation:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372128.image4.png\" alt=\"image4.png\" width=\"251\" height=\"40\" />\r\n<p class=\"child-para\">Follow the order of operations to get this answer:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/372129.image5.png\" alt=\"image5.png\" width=\"295\" height=\"45\" />\r\n<p class=\"child-para\">This equation simplifies to cos(A + B) = 4/5.</p>\r\n</li>\r\n</ol>","blurb":"","authors":[{"authorId":9703,"name":"Yang Kuang","slug":"yang-kuang","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9703"}},{"authorId":9704,"name":"Elleyne Kase","slug":"elleyne-kase","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9704"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-632ca2decefc3\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-632ca2decf877\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-09-22T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":167376},{"headers":{"creationTime":"2016-03-26T15:13:15+00:00","modifiedTime":"2022-08-08T17:50:36+00:00","timestamp":"2022-09-14T18:19:52+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"A Quick Guide to the 30-60-90 Triangle","strippedTitle":"a quick guide to the 30-60-90 triangle","slug":"quick-guide-to-the-30-60-90-degree-triangle","canonicalUrl":"","seo":{"metaDescription":"If you know any one side of a 30-60-90 triangle, you can find the other two. Each of these triangles have many things in common.","noIndex":0,"noFollow":0},"content":"The 30-60-90 triangle is shaped like half of an equilateral triangle, cut straight down the middle along its altitude. It has angles of 30 degrees, 60 degrees, and 90 degrees, thus, its name! In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3.\r\n<p class=\"Tip\">The hypotenuse is the longest side in a right triangle, which is different from the long leg. The long leg is the leg opposite the 60-degree angle.</p>\r\nTwo of the most common right triangles are 30-60-90 and the <a href=\"https://www.dummies.com/article/academics-the-arts/math/pre-calculus/how-to-work-with-45-45-90-degree-triangles-167436/\" target=\"_blank\" rel=\"noopener\">45-45-90-degree triangles</a>. All 30-60-90 triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in <a href=\"https://www.dummies.com/article/academics-the-arts/math/trigonometry/defining-the-radian-in-trigonometry-199411/\" target=\"_blank\" rel=\"noopener\">radians</a>, it translates to the following:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/369579.image0.png\" alt=\"30, 60, and 90 degrees expressed in radians.\" width=\"73\" height=\"37\" />\r\n\r\nThe figure illustrates the ratio of the sides for the 30-60-90-degree triangle.\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/369580.image1.jpg\" alt=\"A 30-60-90-degree right triangle.\" width=\"400\" height=\"257\" /> A 30-60-90-degree right triangle[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div>\r\nIf you know one side of a 30-60-90 triangle, you can find the other two by using shortcuts. Here are the three situations you come across when doing these calculations:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Type 1: You know the short leg (the side across from the 30-degree angle). </b>Double its length to find the hypotenuse. You can multiply the short side by the square root of 3 to find the long leg.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Type 2: You know the hypotenuse. </b>Divide the hypotenuse by 2 to find the short side. Multiply this answer by the square root of 3 to find the long leg.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Type 3: You know the long leg (the side across from the 60-degree angle).</b> Divide this side by the square root of 3 to find the short side. Double that figure to find the hypotenuse.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/369581.image2.jpg\" alt=\"Finding the other sides of a 30-60-90 triangle when you know the hypotenuse.\" width=\"400\" height=\"249\" /> Finding the other sides of a 30-60-90 triangle when you know the hypotenuse[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div></li>\r\n</ul>\r\nIn the triangle TRI in this figure, the hypotenuse is 14 inches long; how long are the other sides?\r\n\r\nBecause you have the hypotenuse TR = 14, you can divide by 2 to get the short side: RI = 7. Now you multiply this length by the square root of 3 to get the long side:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/369582.image3.png\" alt=\"The long side of a 30-60-90-degree triangle.\" width=\"65\" height=\"23\" />","description":"The 30-60-90 triangle is shaped like half of an equilateral triangle, cut straight down the middle along its altitude. It has angles of 30 degrees, 60 degrees, and 90 degrees, thus, its name! In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3.\r\n<p class=\"Tip\">The hypotenuse is the longest side in a right triangle, which is different from the long leg. The long leg is the leg opposite the 60-degree angle.</p>\r\nTwo of the most common right triangles are 30-60-90 and the <a href=\"https://www.dummies.com/article/academics-the-arts/math/pre-calculus/how-to-work-with-45-45-90-degree-triangles-167436/\" target=\"_blank\" rel=\"noopener\">45-45-90-degree triangles</a>. All 30-60-90 triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in <a href=\"https://www.dummies.com/article/academics-the-arts/math/trigonometry/defining-the-radian-in-trigonometry-199411/\" target=\"_blank\" rel=\"noopener\">radians</a>, it translates to the following:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/369579.image0.png\" alt=\"30, 60, and 90 degrees expressed in radians.\" width=\"73\" height=\"37\" />\r\n\r\nThe figure illustrates the ratio of the sides for the 30-60-90-degree triangle.\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/369580.image1.jpg\" alt=\"A 30-60-90-degree right triangle.\" width=\"400\" height=\"257\" /> A 30-60-90-degree right triangle[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div>\r\nIf you know one side of a 30-60-90 triangle, you can find the other two by using shortcuts. Here are the three situations you come across when doing these calculations:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Type 1: You know the short leg (the side across from the 30-degree angle). </b>Double its length to find the hypotenuse. You can multiply the short side by the square root of 3 to find the long leg.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Type 2: You know the hypotenuse. </b>Divide the hypotenuse by 2 to find the short side. Multiply this answer by the square root of 3 to find the long leg.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Type 3: You know the long leg (the side across from the 60-degree angle).</b> Divide this side by the square root of 3 to find the short side. Double that figure to find the hypotenuse.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/369581.image2.jpg\" alt=\"Finding the other sides of a 30-60-90 triangle when you know the hypotenuse.\" width=\"400\" height=\"249\" /> Finding the other sides of a 30-60-90 triangle when you know the hypotenuse[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div></li>\r\n</ul>\r\nIn the triangle TRI in this figure, the hypotenuse is 14 inches long; how long are the other sides?\r\n\r\nBecause you have the hypotenuse TR = 14, you can divide by 2 to get the short side: RI = 7. Now you multiply this length by the square root of 3 to get the long side:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/369582.image3.png\" alt=\"The long side of a 30-60-90-degree triangle.\" width=\"65\" height=\"23\" />","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}},{"articleId":260215,"title":"The Differences between Pre-Calculus and Calculus","slug":"the-differences-between-pre-calculus-and-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260215"}},{"articleId":260207,"title":"10 Polar Graphs","slug":"10-polar-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260207"}},{"articleId":260183,"title":"Pre-Calculus: 10 Habits to Adjust before Calculus","slug":"pre-calculus-10-habits-to-adjust-before-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260183"}},{"articleId":208308,"title":"Pre-Calculus For Dummies Cheat Sheet","slug":"pre-calculus-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208308"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282496,"slug":"pre-calculus-for-dummies-3rd-edition","isbn":"9781119508779","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508770-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/pre-calculus-for-dummies-3rd-edition-cover-9781119508779-203x255.jpg","width":203,"height":255},"title":"Pre-Calculus For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. She is the author of several <i>For Dummies books,</i> including <i>Algebra Workbook For Dummies, Algebra II For Dummies,</i> and <i>Algebra II Workbook For Dummies.</i> </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508779&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;1119508770&quot;]},{&quot;key&quot;:&quot;test&quot;,&quot;values&quot;:[&quot;control1564&quot;]}]\" id=\"du-slot-63221b482725d\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508779&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;1119508770&quot;]},{&quot;key&quot;:&quot;test&quot;,&quot;values&quot;:[&quot;control1564&quot;]}]\" id=\"du-slot-63221b4827c59\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-08T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[{"adPairKey":"isbn","adPairValue":"1119508770"},{"adPairKey":"test","adPairValue":"control1564"}]},"status":"publish","visibility":"public","articleId":168056},{"headers":{"creationTime":"2016-03-27T16:54:23+00:00","modifiedTime":"2022-04-06T16:45:44+00:00","timestamp":"2022-09-14T18:19:34+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Pre-Calculus Workbook For Dummies Cheat Sheet","strippedTitle":"pre-calculus workbook for dummies cheat sheet","slug":"pre-calculus-workbook-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Prepare for calculus by learning formulas, laws of logarithmic functions, trigonometric values of basic angles, and interval notation.","noIndex":0,"noFollow":0},"content":"Pre-calculus uses the information you know from Algebra I and II and ratchets up the difficulty level to prepare you for calculus. This cheat sheet is designed to help you review key formulas and functions on the fly as you study. It includes formulas, the laws of logarithmic functions, trigonometric values of basic angles, conic section equations, and interval notation.","description":"Pre-calculus uses the information you know from Algebra I and II and ratchets up the difficulty level to prepare you for calculus. This cheat sheet is designed to help you review key formulas and functions on the fly as you study. It includes formulas, the laws of logarithmic functions, trigonometric values of basic angles, conic section equations, and interval notation.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":167923,"title":"How to Find Imaginary Roots Using the Fundamental Theorem of Algebra","slug":"how-to-find-imaginary-roots-using-the-fundamental-theorem-of-algebra","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167923"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282497,"slug":"pre-calculus-workbook-for-dummies-3rd-edition","isbn":"9781119508809","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508800-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/pre-calculus-workbook-for-dummies-3rd-edition-cover-9781119508809-204x255.jpg","width":204,"height":255},"title":"Pre-Calculus Workbook For Dummies","testBankPinActivationLink":"https://testbanks.wiley.com","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of <i>Trigonometry For Dummies</i> and <i>Finite Math For Dummies.</i> </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b36cf1b4\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b36cfba2\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":260274,"title":"Pre-Calculus Formulas","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260274"}},{"articleId":260300,"title":"How to Identify Conic Section Equations in Pre-Calculus","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260300"}},{"articleId":260303,"title":"Laws of Exponents and Logarithms for Pre-Calculus","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260303"}},{"articleId":260306,"title":"Trigonometric Values of Basic Angles for Pre-Calculus","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260306"}},{"articleId":260309,"title":"Interval Notation in Pre-Calculus","slug":"","categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260309"}}],"content":[{"title":"Pre-Calculus formulas","thumb":null,"image":null,"content":"<p>Pre-calculus requires a thorough understanding of the topics covered in Algebra I and Algebra II, along with some of the basic concepts of calculus. In solving pre-calculus problems, a number of formulas come in handy. The following list collects some of the more difficult-to-remember formulas for your convenience and for ease of study.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-260295\" src=\"https://www.dummies.com/wp-content/uploads/pre-calculus-formulas.jpg\" alt=\"pre-calculus-formulas\" width=\"439\" height=\"306\" /></p>\n"},{"title":"How to identify conic section equations in pre-calculus","thumb":null,"image":null,"content":"<p>Differentiating among the equations for conic sections is a challenging part of pre-calculus. Understanding the differences between parabolas and hyperbolas, circles and ellipses, and so on, can be tricky. The following list explains how the equations for each are unique:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-260293\" src=\"https://www.dummies.com/wp-content/uploads/pre-calculus-conic-sections.jpg\" alt=\"pre-calculus-conic-sections\" width=\"456\" height=\"239\" /></p>\n"},{"title":"Laws of exponents and logarithms for pre-calculus","thumb":null,"image":null,"content":"<p>Among the trickier aspects of pre-calculus for beginners to remember are the rules for exponents and logarithms. <em>Exponents</em>, of course, indicate the operation of raising a number to a power, or in other words, of multiplying a number by itself. <em>Logarithms</em> are simply another way to write exponents: Exponential and logarithmic functions are inverses of each other. For solving and graphing logarithmic functions (logs), remember this inverse relationship and you’ll be solving logs in no time! The following lists describe the basic rules for each.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-260294\" src=\"https://www.dummies.com/wp-content/uploads/pre-calculus-exponents-logarithms.jpg\" alt=\"pre-calculus-exponents-logarithms\" width=\"288\" height=\"445\" /></p>\n"},{"title":"Trigonometric values of basic angles for pre-calculus","thumb":null,"image":null,"content":"<p>Of course you use trigonometry, commonly called trig, in pre-calculus. And understanding the trig values of common angles is an important part of solving trig problems. The following table shows the basic values for each.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-260297\" src=\"https://www.dummies.com/wp-content/uploads/pre-calculus-trigonometric-values.jpg\" alt=\"pre-calculus-trigonometric-values\" width=\"227\" height=\"301\" /></p>\n"},{"title":"Interval notation in pre-calculus","thumb":null,"image":null,"content":"<p>In pre-calculus you deal with inequalities, and you use interval notation to express a solution set to an inequality. The following list shows how to format solution sets in interval notation.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-260296\" src=\"https://www.dummies.com/wp-content/uploads/pre-calculus-interval-notation.jpg\" alt=\"pre-calculus-interval-notation\" width=\"164\" height=\"217\" /></p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-04-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":208683},{"headers":{"creationTime":"2016-03-27T16:47:52+00:00","modifiedTime":"2022-02-24T19:58:32+00:00","timestamp":"2022-09-14T18:19:14+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Pre-Calculus: 1001 Practice Problems For Dummies Cheat Sheet","strippedTitle":"pre-calculus: 1001 practice problems for dummies cheat sheet","slug":"1001-pre-calculus-practice-problems-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This helpful Cheat Sheet is a quick reference for what you should know as you move into the study of calculus.","noIndex":0,"noFollow":0},"content":"Pre-calculus draws from algebra, geometry, and trigonometry and combines these topics to prepare you for the techniques you need to succeed in calculus.\r\n\r\nThis cheat sheet provides the most frequently used formulas, with brief descriptions of what the letters and symbols represent. Counting techniques are also here, letting you count numbers of events without actually having to list all the ways to do them. Also, you find a step-by-step description of how to <i>complete the square</i> — most useful when you’re working with conic sections and other equations with specific formats.","description":"Pre-calculus draws from algebra, geometry, and trigonometry and combines these topics to prepare you for the techniques you need to succeed in calculus.\r\n\r\nThis cheat sheet provides the most frequently used formulas, with brief descriptions of what the letters and symbols represent. Counting techniques are also here, letting you count numbers of events without actually having to list all the ways to do them. Also, you find a step-by-step description of how to <i>complete the square</i> — most useful when you’re working with conic sections and other equations with specific formats.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":149120,"title":"Counting Techniques for Pre-Calculus","slug":"counting-techniques-for-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149120"}},{"articleId":149116,"title":"Completing the Square for Pre-Calculus","slug":"completing-the-square-for-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149116"}},{"articleId":149117,"title":"Frequently Used Pre-Calculus Formulas","slug":"frequently-used-pre-calculus-formulas","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149117"}},{"articleId":146760,"title":"Algebra Basics Needed for Pre-Calculus","slug":"algebra-basics-needed-for-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/146760"}},{"articleId":146758,"title":"Complex Numbers in Pre-Calculus","slug":"complex-numbers-in-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/146758"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282498,"slug":"pre-calculus-for-dummies-1001-practice-problems","isbn":"9781119883623","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119883628-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/9781119883623-204x255.jpg","width":204,"height":255},"title":"Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"8985\">Mary Jane Sterling</b></b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883623&quot;]}]\" id=\"du-slot-63221b2244c37\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883623&quot;]}]\" id=\"du-slot-63221b2245550\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":149117,"title":"Frequently Used Pre-Calculus Formulas","slug":"frequently-used-pre-calculus-formulas","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149117"}},{"articleId":149120,"title":"Counting Techniques for Pre-Calculus","slug":"counting-techniques-for-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149120"}},{"articleId":149116,"title":"Completing the Square for Pre-Calculus","slug":"completing-the-square-for-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149116"}}],"content":[{"title":"Frequently used pre-calculus formulas","thumb":null,"image":null,"content":"<p><i>Mathematical formulas</i> are equations that are always true. You can use them in algebra, geometry, trigonometry, and many other mathematical applications, including pre-calculus. Refer to these formulas when you need a quick reminder of exactly what those equations are and which measures or inputs are needed.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/440933.image0.png\" alt=\"image0.png\" width=\"535\" height=\"421\" /></p>\n"},{"title":"Counting techniques for pre-calculus","thumb":null,"image":null,"content":"<p>Counting the number of ways to perform a task is fairly simple — until the number of choices gets too large. Here are three counting techniques used in pre-calculus:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Permutations:</b> How many ways you can choose <i>r</i> things from a total of <i>n</i> things when the order of your choices matters</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Combinations:</b> How many ways you can choose <i>r</i> things from a total of <i>n</i> things when the order of your choices doesn’t matter</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Multiplication principle:</b> How many different possibilities exist if you choose 1 from <i>a </i>things, 1 from <i>b</i> things, 1 from <i>c</i> things, 1 from <i>d</i> things, and so on.</p>\n</li>\n</ul>\n<p>Use the following formulas for these counting techniques:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/440935.image0.png\" alt=\"image0.png\" width=\"206\" height=\"140\" /></p>\n"},{"title":"Completing the square for conic sections","thumb":null,"image":null,"content":"<p>When the equation of a conic section isn&#8217;t written in its standard form, completing the square is the only way to convert the equation to its standard form. The steps of the process are as follows:</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">Add/subtract any constant to the opposite side of the given equation, away from all the variables.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor the leading coefficient out of all terms in front of the set of parentheses.</p>\n</li>\n<li>\n<p class=\"first-para\">Divide the remaining linear coefficient by two, but only in your head.</p>\n</li>\n<li>\n<p class=\"first-para\">Square the answer from Step 3 and add that inside the parentheses.</p>\n<p class=\"child-para\">Don&#8217;t forget that if you have a coefficient from Step 2, you must multiply the coefficient by the number you get in this step and add <i>that</i> to both sides.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor the quadratic polynomial as a perfect square trinomial.</p>\n</li>\n</ol>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-01-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207642},{"headers":{"creationTime":"2016-03-26T15:11:54+00:00","modifiedTime":"2021-12-21T21:22:41+00:00","timestamp":"2022-09-14T18:18:56+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"How to Find Imaginary Roots Using the Fundamental Theorem of Algebra","strippedTitle":"how to find imaginary roots using the fundamental theorem of algebra","slug":"how-to-find-imaginary-roots-using-the-fundamental-theorem-of-algebra","canonicalUrl":"","seo":{"metaDescription":"Learn about the fundamental theorem of algebra, what imaginary roots are, and why the quadratic formula always gives you two solutions.","noIndex":0,"noFollow":0},"content":"The fundamental theorem of algebra can help you find imaginary roots. <i>Imaginary roots</i> appear in a quadratic equation when the discriminant of the quadratic equation — the part under the square root sign (<i>b</i><sup>2</sup> – 4<i>ac</i>) — is negative. If this value is negative, you can’t actually take the square root, and the answers are not real. In other words, there is no real solution; therefore, the graph won’t cross the <i>x-</i>axis.\r\n\r\nUsing the quadratic formula always gives you two solutions, because the plus/minus sign means you’re both adding and subtracting and getting two completely different answers. When the number underneath the square-root sign in the quadratic formula is negative, the answers are called <i>complex conjugates.</i> One is <i>r</i> + <i>si</i> and the other is <i>r</i> – <i>si.</i> These numbers have both real (the <i>r</i>) and imaginary (the <i>si</i>) parts.\r\n<p class=\"TechnicalStuff\">The complex number system consists of all numbers <i>r + si</i> where <i>r</i> and <i>s</i> are real numbers. Observe that when <i>s = 0</i>, you simply have the real numbers. Therefore the real numbers are a subset of the complex number system. <i>The fundamental theorem of algebra</i> says that every polynomial function has at least one root in the complex number system.</p>\r\nThe highest degree of a polynomial gives you the highest possible number of distinct <i>complex </i>roots for the polynomial. Between this fact and Descartes’s rule of signs, you can get an idea of how many imaginary roots a polynomial has.\r\n\r\nHere’s how Descartes’s rule of signs can give you the numbers of possible real roots, both positive and negative:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Positive real roots.</b> For the number of positive real roots, look at the polynomial, written in descending order, and count how many times the sign changes from term to term. This value represents the maximum number of positive roots in the polynomial. For example, in the polynomial <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48, you see two changes in sign (don’t forget to include the sign of the first term!) — from the first term (+<i>2x</i><i><sup>4</sup></i>) to the second (-<i>9x</i><i><sup>3</sup></i>) and from the third term (-<i>21x</i><i><sup>2</sup></i>) to the fourth term (<i>88x</i>). That means this equation can have up to two positive solutions.</p>\r\n<p class=\"child-para\"><i></i>Descartes’s rule of signs says the number of positive roots is equal to changes in sign of <i>f</i>(<i>x</i>), or is less than that by an even number (so you keep subtracting 2 until you get either 1 or 0). Therefore, the previous <i>f</i>(<i>x</i>) may have 2 or 0 positive roots.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Negative real roots. </b>For the number of negative real roots, find <i>f</i>(–<i>x</i>) and count again. Because negative numbers raised to even powers are positive and negative numbers raised to odd powers are negative, this change affects only terms with odd powers. This step is the same as changing each term with an odd degree to its opposite sign and counting the sign changes again, which gives you the maximum number of negative roots. The example equation becomes <i>f</i>(–<i>x</i>) = 2<i>x</i><sup>4</sup> + 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> – 88<i>x </i>+ 48, which changes signs twice. There can be, at most, two negative roots.<i> </i>However, similar to the rule for positive roots, the number of negative roots is equal to the changes in sign for <i>f</i>(–<i>x</i>), or must be less than that by an even number. Therefore, this example can have either 2 or 0 negative roots.</p>\r\n</li>\r\n</ul>\r\nPair up every possible number of positive real roots with every possible number of negative real roots; the remaining number of roots for each situation represents the number of imaginary roots.\r\n\r\nFor example, the polynomial <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48 has a degree of 4, with two or zero positive real roots, and two or zero negative real roots. With this information, you can pair up the possible situations:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Two positive and two negative real roots, with zero imaginary roots</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Two positive and zero negative real roots, with two imaginary roots</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Zero positive and two negative real roots, with two imaginary roots</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Zero positive and zero negative real roots, with four imaginary roots</p>\r\n</li>\r\n</ul>\r\nThe following chart makes the information easier to picture:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Positive real roots</th>\r\n<th>Negative real roots</th>\r\n<th>Imaginary roots</th>\r\n</tr>\r\n<tr>\r\n<td>2</td>\r\n<td>2</td>\r\n<td>0</td>\r\n</tr>\r\n<tr>\r\n<td>2</td>\r\n<td>0</td>\r\n<td>2</td>\r\n</tr>\r\n<tr>\r\n<td>0</td>\r\n<td>2</td>\r\n<td>2</td>\r\n</tr>\r\n<tr>\r\n<td>0</td>\r\n<td>0</td>\r\n<td>4</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\nComplex numbers are written in the form <i>r</i> + <i>si</i> and have both a real and an imaginary part, which is why every polynomial has at least one root in the complex number system. Real and imaginary numbers are both included in the complex number system. Real numbers have no imaginary part, and pure imaginary numbers have no real part. For example, if <i>x</i> = 7 is one root of the polynomial, this root is considered both real and complex because it can be rewritten as <i>x</i> = 7 + 0<i>i</i> (the imaginary part is 0).\r\n\r\nThe fundamental theorem of algebra gives the total number of complex roots (say there are seven); Descartes’s rule of signs tells you how many possible real roots exist and how many of them are positive and negative (say<i> </i>there are, at most, two positive roots but only one negative root). Now, assume you’ve found them all: <i>x</i> = 1, <i>x</i> = 7, and <i>x</i> = –2. These roots are real, but they’re also complex because they can all be rewritten.\r\n\r\nThe first two columns in the chart find the real roots and classify them as positive or negative. The third column is actually finding, specifically, the non-real numbers: complex numbers with non-zero imaginary parts.","description":"The fundamental theorem of algebra can help you find imaginary roots. <i>Imaginary roots</i> appear in a quadratic equation when the discriminant of the quadratic equation — the part under the square root sign (<i>b</i><sup>2</sup> – 4<i>ac</i>) — is negative. If this value is negative, you can’t actually take the square root, and the answers are not real. In other words, there is no real solution; therefore, the graph won’t cross the <i>x-</i>axis.\r\n\r\nUsing the quadratic formula always gives you two solutions, because the plus/minus sign means you’re both adding and subtracting and getting two completely different answers. When the number underneath the square-root sign in the quadratic formula is negative, the answers are called <i>complex conjugates.</i> One is <i>r</i> + <i>si</i> and the other is <i>r</i> – <i>si.</i> These numbers have both real (the <i>r</i>) and imaginary (the <i>si</i>) parts.\r\n<p class=\"TechnicalStuff\">The complex number system consists of all numbers <i>r + si</i> where <i>r</i> and <i>s</i> are real numbers. Observe that when <i>s = 0</i>, you simply have the real numbers. Therefore the real numbers are a subset of the complex number system. <i>The fundamental theorem of algebra</i> says that every polynomial function has at least one root in the complex number system.</p>\r\nThe highest degree of a polynomial gives you the highest possible number of distinct <i>complex </i>roots for the polynomial. Between this fact and Descartes’s rule of signs, you can get an idea of how many imaginary roots a polynomial has.\r\n\r\nHere’s how Descartes’s rule of signs can give you the numbers of possible real roots, both positive and negative:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Positive real roots.</b> For the number of positive real roots, look at the polynomial, written in descending order, and count how many times the sign changes from term to term. This value represents the maximum number of positive roots in the polynomial. For example, in the polynomial <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48, you see two changes in sign (don’t forget to include the sign of the first term!) — from the first term (+<i>2x</i><i><sup>4</sup></i>) to the second (-<i>9x</i><i><sup>3</sup></i>) and from the third term (-<i>21x</i><i><sup>2</sup></i>) to the fourth term (<i>88x</i>). That means this equation can have up to two positive solutions.</p>\r\n<p class=\"child-para\"><i></i>Descartes’s rule of signs says the number of positive roots is equal to changes in sign of <i>f</i>(<i>x</i>), or is less than that by an even number (so you keep subtracting 2 until you get either 1 or 0). Therefore, the previous <i>f</i>(<i>x</i>) may have 2 or 0 positive roots.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Negative real roots. </b>For the number of negative real roots, find <i>f</i>(–<i>x</i>) and count again. Because negative numbers raised to even powers are positive and negative numbers raised to odd powers are negative, this change affects only terms with odd powers. This step is the same as changing each term with an odd degree to its opposite sign and counting the sign changes again, which gives you the maximum number of negative roots. The example equation becomes <i>f</i>(–<i>x</i>) = 2<i>x</i><sup>4</sup> + 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> – 88<i>x </i>+ 48, which changes signs twice. There can be, at most, two negative roots.<i> </i>However, similar to the rule for positive roots, the number of negative roots is equal to the changes in sign for <i>f</i>(–<i>x</i>), or must be less than that by an even number. Therefore, this example can have either 2 or 0 negative roots.</p>\r\n</li>\r\n</ul>\r\nPair up every possible number of positive real roots with every possible number of negative real roots; the remaining number of roots for each situation represents the number of imaginary roots.\r\n\r\nFor example, the polynomial <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48 has a degree of 4, with two or zero positive real roots, and two or zero negative real roots. With this information, you can pair up the possible situations:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Two positive and two negative real roots, with zero imaginary roots</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Two positive and zero negative real roots, with two imaginary roots</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Zero positive and two negative real roots, with two imaginary roots</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Zero positive and zero negative real roots, with four imaginary roots</p>\r\n</li>\r\n</ul>\r\nThe following chart makes the information easier to picture:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Positive real roots</th>\r\n<th>Negative real roots</th>\r\n<th>Imaginary roots</th>\r\n</tr>\r\n<tr>\r\n<td>2</td>\r\n<td>2</td>\r\n<td>0</td>\r\n</tr>\r\n<tr>\r\n<td>2</td>\r\n<td>0</td>\r\n<td>2</td>\r\n</tr>\r\n<tr>\r\n<td>0</td>\r\n<td>2</td>\r\n<td>2</td>\r\n</tr>\r\n<tr>\r\n<td>0</td>\r\n<td>0</td>\r\n<td>4</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\nComplex numbers are written in the form <i>r</i> + <i>si</i> and have both a real and an imaginary part, which is why every polynomial has at least one root in the complex number system. Real and imaginary numbers are both included in the complex number system. Real numbers have no imaginary part, and pure imaginary numbers have no real part. For example, if <i>x</i> = 7 is one root of the polynomial, this root is considered both real and complex because it can be rewritten as <i>x</i> = 7 + 0<i>i</i> (the imaginary part is 0).\r\n\r\nThe fundamental theorem of algebra gives the total number of complex roots (say there are seven); Descartes’s rule of signs tells you how many possible real roots exist and how many of them are positive and negative (say<i> </i>there are, at most, two positive roots but only one negative root). Now, assume you’ve found them all: <i>x</i> = 1, <i>x</i> = 7, and <i>x</i> = –2. These roots are real, but they’re also complex because they can all be rewritten.\r\n\r\nThe first two columns in the chart find the real roots and classify them as positive or negative. The third column is actually finding, specifically, the non-real numbers: complex numbers with non-zero imaginary parts.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":208683,"title":"Pre-Calculus Workbook For Dummies Cheat Sheet","slug":"pre-calculus-workbook-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208683"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282497,"slug":"pre-calculus-workbook-for-dummies-3rd-edition","isbn":"9781119508809","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508800-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/pre-calculus-workbook-for-dummies-3rd-edition-cover-9781119508809-204x255.jpg","width":204,"height":255},"title":"Pre-Calculus Workbook For Dummies","testBankPinActivationLink":"https://testbanks.wiley.com","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of <i>Trigonometry For Dummies</i> and <i>Finite Math For Dummies.</i> </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b1081c77\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b10826a3\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":167923},{"headers":{"creationTime":"2016-03-26T15:07:27+00:00","modifiedTime":"2021-12-21T21:13:55+00:00","timestamp":"2022-09-14T18:18:56+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"How to Graph Polar Coordinates with Negative Values","strippedTitle":"how to graph polar coordinates with negative values","slug":"how-to-graph-polar-coordinates-with-negative-values","canonicalUrl":"","seo":{"metaDescription":"This article provides a step-by-step guide to graphing polar coordinates with negative angles and/or radii.","noIndex":0,"noFollow":0},"content":"Sometimes your geometry teacher may spice things up a bit with <em>complicated polar coordinates</em> — points with negative angles and/or radii. The following list shows you how to plot in three situations — when the angle is negative, when the radius is negative, and when both are negative.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><strong>When the angle is negative: </strong>Negative angles move in a clockwise direction.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 600px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"450\"]<img src=\"https://www.dummies.com/wp-content/uploads/371605.image0.png\" alt=\"Visualizing simple and complex polar coordinates.\" width=\"450\" height=\"451\" /> Visualizing simple and complex polar coordinates[/caption]\r\n\r\n</div>\r\n<p class=\"child-para\">This figure shows an example point, D. To locate the polar coordinate point D at</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371606.image1.png\" alt=\"image1.png\" width=\"63\" height=\"40\" />\r\n<p class=\"child-para\">first locate the angle</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371607.image2.png\" alt=\"image2.png\" width=\"25\" height=\"37\" />\r\n<p class=\"child-para\">and then find the location of the radius, 1, on that line.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>When the radius is negative:</strong> When graphing a polar coordinate with a negative radius, you move from the pole in the direction opposite the given positive angle (on the same line as the given angle but in the direction opposite to the angle from the pole). For example, check out point F at</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371608.image3.png\" alt=\"image3.png\" width=\"63\" height=\"40\" />\r\n<p class=\"child-para\">in the figure.</p>\r\n<p class=\"child-para Tip\">Some teachers prefer to teach their students to move right along the <em>x-</em> (polar) axis for positive numbers (radii) and left for negative. Then you do the rotation for the angle in a positive direction. You’ll get to the same spot with that method.</p>\r\n<p class=\"child-para\">For example, take a look at point F</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371609.image4.png\" alt=\"image4.png\" width=\"63\" height=\"40\" />\r\n<p class=\"child-para\">in the figure. Because the radius is negative, move along the left <em>x-</em>axis 1/2 of a unit. Then rotate the angle in the positive direction (counterclockwise) pi/3 radians. You should arrive at your destination, point F.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>When both the angle and radius are negative: </strong>To express a polar coordinate with a negative radius and a negative angle, locate the terminal side of the negative angle first and then move in the opposite direction to locate the radius. For example, point G in the figure has these characteristics at</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371610.image5.png\" alt=\"image5.png\" width=\"83\" height=\"40\" /></li>\r\n</ul>\r\nIndeed, except the origin, each given point can have the following four types of representations:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Positive radius, positive angle</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Positive radius, negative angle</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Negative radius, positive angle</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Negative radius, negative angle</p>\r\n</li>\r\n</ul>\r\nFor example, point E in the figure\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371611.image6.png\" alt=\"image6.png\" width=\"49\" height=\"40\" />\r\n\r\ncan have three other polar coordinate representations with different combinations of signs for the radius and angle:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371612.image7.png\" alt=\"image7.png\" width=\"79\" height=\"120\" />\r\n\r\nWhen polar graphing, you can change the coordinate of any point you’re given into polar coordinates that are easy to deal with (such as positive radius, positive angle).","description":"Sometimes your geometry teacher may spice things up a bit with <em>complicated polar coordinates</em> — points with negative angles and/or radii. The following list shows you how to plot in three situations — when the angle is negative, when the radius is negative, and when both are negative.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><strong>When the angle is negative: </strong>Negative angles move in a clockwise direction.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 600px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"450\"]<img src=\"https://www.dummies.com/wp-content/uploads/371605.image0.png\" alt=\"Visualizing simple and complex polar coordinates.\" width=\"450\" height=\"451\" /> Visualizing simple and complex polar coordinates[/caption]\r\n\r\n</div>\r\n<p class=\"child-para\">This figure shows an example point, D. To locate the polar coordinate point D at</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371606.image1.png\" alt=\"image1.png\" width=\"63\" height=\"40\" />\r\n<p class=\"child-para\">first locate the angle</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371607.image2.png\" alt=\"image2.png\" width=\"25\" height=\"37\" />\r\n<p class=\"child-para\">and then find the location of the radius, 1, on that line.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>When the radius is negative:</strong> When graphing a polar coordinate with a negative radius, you move from the pole in the direction opposite the given positive angle (on the same line as the given angle but in the direction opposite to the angle from the pole). For example, check out point F at</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371608.image3.png\" alt=\"image3.png\" width=\"63\" height=\"40\" />\r\n<p class=\"child-para\">in the figure.</p>\r\n<p class=\"child-para Tip\">Some teachers prefer to teach their students to move right along the <em>x-</em> (polar) axis for positive numbers (radii) and left for negative. Then you do the rotation for the angle in a positive direction. You’ll get to the same spot with that method.</p>\r\n<p class=\"child-para\">For example, take a look at point F</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371609.image4.png\" alt=\"image4.png\" width=\"63\" height=\"40\" />\r\n<p class=\"child-para\">in the figure. Because the radius is negative, move along the left <em>x-</em>axis 1/2 of a unit. Then rotate the angle in the positive direction (counterclockwise) pi/3 radians. You should arrive at your destination, point F.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>When both the angle and radius are negative: </strong>To express a polar coordinate with a negative radius and a negative angle, locate the terminal side of the negative angle first and then move in the opposite direction to locate the radius. For example, point G in the figure has these characteristics at</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371610.image5.png\" alt=\"image5.png\" width=\"83\" height=\"40\" /></li>\r\n</ul>\r\nIndeed, except the origin, each given point can have the following four types of representations:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Positive radius, positive angle</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Positive radius, negative angle</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Negative radius, positive angle</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Negative radius, negative angle</p>\r\n</li>\r\n</ul>\r\nFor example, point E in the figure\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371611.image6.png\" alt=\"image6.png\" width=\"49\" height=\"40\" />\r\n\r\ncan have three other polar coordinate representations with different combinations of signs for the radius and angle:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/371612.image7.png\" alt=\"image7.png\" width=\"79\" height=\"120\" />\r\n\r\nWhen polar graphing, you can change the coordinate of any point you’re given into polar coordinates that are easy to deal with (such as positive radius, positive angle).","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":208683,"title":"Pre-Calculus Workbook For Dummies Cheat Sheet","slug":"pre-calculus-workbook-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208683"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282497,"slug":"pre-calculus-workbook-for-dummies-3rd-edition","isbn":"9781119508809","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508800-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/pre-calculus-workbook-for-dummies-3rd-edition-cover-9781119508809-204x255.jpg","width":204,"height":255},"title":"Pre-Calculus Workbook For Dummies","testBankPinActivationLink":"https://testbanks.wiley.com","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of <i>Trigonometry For Dummies</i> and <i>Finite Math For Dummies.</i> </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b1078b63\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b10795a1\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-13T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":167448},{"headers":{"creationTime":"2016-03-26T15:10:44+00:00","modifiedTime":"2021-12-21T20:52:30+00:00","timestamp":"2022-09-14T18:18:56+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"How to Find Binomial Coefficients","strippedTitle":"how to find binomial coefficients","slug":"how-to-find-binomial-coefficients","canonicalUrl":"","seo":{"metaDescription":"In calculus, you can use a shortcut called Pascal's triangle to find binomial coefficients, if the exponent is relative small.","noIndex":0,"noFollow":0},"content":"Depending on how many times you must multiply the same binomial — a value also known as an <i>exponent</i> — the binomial coefficients for that particular exponent are always the same. The binomial coefficients are found by using the\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370761.image0.png\" alt=\"image0.png\" width=\"44\" height=\"48\" />\r\n\r\ncombinations formula. If the exponent is relatively small, you can use a shortcut called <i>Pascal</i><i>'</i><i>s triangle </i>to find these coefficients. If not, you can always rely on algebra!\r\n\r\n<i>Pascal</i><i>'</i><i>s triangle,</i> named after the famous mathematician Blaise Pascal, names the binomial coefficients for the binomial expansion. It is especially useful when raising a binomial to lower degrees.\r\n\r\nFor example, if a sadistic teacher asked you to find (3<i>x</i> + 4)<sup>10</sup>, you probably wouldn't want to use Pascal's triangle; instead, you'd just use the algebraic formula described shortly. The figure illustrates this concept. The top number of the triangle is 1, as well as all the numbers on the outer sides. To get any term in the triangle, you find the sum of the two numbers above it.\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/370762.image1.jpg\" alt=\"Determining coefficients with Pascal's triangle.\" width=\"400\" height=\"227\" /> Determining coefficients with Pascal's triangle[/caption]\r\n\r\n</div>\r\nEach row gives the coefficients to (<i>a</i> + <i>b</i>)<i><sup>n</sup></i>, starting with <i>n</i> = 0. To find the binomial coefficients for (<i>a</i> + <i>b</i>)<i><sup>n</sup></i>, use the <i>n</i>th row and always start with the beginning. For instance, the binomial coefficients for (<i>a</i> + <i>b</i>)<sup>5</sup> are 1, 5, 10, 10, 5, and 1 — in that order.\r\n\r\nIf you need to find the coefficients of binomials algebraically, there is a formula for that as well. The <i>r</i>th coefficient for the <i>n</i>th binomial expansion is written in the following form:\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370763.image2.png\" alt=\"image2.png\" width=\"115\" height=\"49\" />\r\n\r\nYou may recall the term <i>factorial</i> from your earlier math classes. If not, here is a reminder: <i>n</i>!, which reads as \"n factorial,\" is defined as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370764.image3.png\" alt=\"image3.png\" width=\"189\" height=\"27\" />\r\n\r\nYou read the expression for the binomial coefficient\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370765.image4.png\" alt=\"image4.png\" width=\"44\" height=\"48\" />\r\n\r\nas \"<i>n</i> choose <i>r.</i>\" You usually can find a button for combinations on a calculator. If not, you can use the factorial button and do each part separately.\r\n\r\nTo make things a little easier, 0! is defined as 1. Therefore, you have these equalities:\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370766.image5.png\" alt=\"image5.png\" width=\"179\" height=\"95\" />\r\n\r\nFor example, to find the binomial coefficient given by\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370767.image6.png\" alt=\"image6.png\" width=\"48\" height=\"48\" />\r\n\r\nsubstitute the values into the formula:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370768.image7.png\" alt=\"image7.png\" width=\"232\" height=\"49\" />","description":"Depending on how many times you must multiply the same binomial — a value also known as an <i>exponent</i> — the binomial coefficients for that particular exponent are always the same. The binomial coefficients are found by using the\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370761.image0.png\" alt=\"image0.png\" width=\"44\" height=\"48\" />\r\n\r\ncombinations formula. If the exponent is relatively small, you can use a shortcut called <i>Pascal</i><i>'</i><i>s triangle </i>to find these coefficients. If not, you can always rely on algebra!\r\n\r\n<i>Pascal</i><i>'</i><i>s triangle,</i> named after the famous mathematician Blaise Pascal, names the binomial coefficients for the binomial expansion. It is especially useful when raising a binomial to lower degrees.\r\n\r\nFor example, if a sadistic teacher asked you to find (3<i>x</i> + 4)<sup>10</sup>, you probably wouldn't want to use Pascal's triangle; instead, you'd just use the algebraic formula described shortly. The figure illustrates this concept. The top number of the triangle is 1, as well as all the numbers on the outer sides. To get any term in the triangle, you find the sum of the two numbers above it.\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"400\"]<img src=\"https://www.dummies.com/wp-content/uploads/370762.image1.jpg\" alt=\"Determining coefficients with Pascal's triangle.\" width=\"400\" height=\"227\" /> Determining coefficients with Pascal's triangle[/caption]\r\n\r\n</div>\r\nEach row gives the coefficients to (<i>a</i> + <i>b</i>)<i><sup>n</sup></i>, starting with <i>n</i> = 0. To find the binomial coefficients for (<i>a</i> + <i>b</i>)<i><sup>n</sup></i>, use the <i>n</i>th row and always start with the beginning. For instance, the binomial coefficients for (<i>a</i> + <i>b</i>)<sup>5</sup> are 1, 5, 10, 10, 5, and 1 — in that order.\r\n\r\nIf you need to find the coefficients of binomials algebraically, there is a formula for that as well. The <i>r</i>th coefficient for the <i>n</i>th binomial expansion is written in the following form:\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370763.image2.png\" alt=\"image2.png\" width=\"115\" height=\"49\" />\r\n\r\nYou may recall the term <i>factorial</i> from your earlier math classes. If not, here is a reminder: <i>n</i>!, which reads as \"n factorial,\" is defined as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370764.image3.png\" alt=\"image3.png\" width=\"189\" height=\"27\" />\r\n\r\nYou read the expression for the binomial coefficient\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370765.image4.png\" alt=\"image4.png\" width=\"44\" height=\"48\" />\r\n\r\nas \"<i>n</i> choose <i>r.</i>\" You usually can find a button for combinations on a calculator. If not, you can use the factorial button and do each part separately.\r\n\r\nTo make things a little easier, 0! is defined as 1. Therefore, you have these equalities:\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370766.image5.png\" alt=\"image5.png\" width=\"179\" height=\"95\" />\r\n\r\nFor example, to find the binomial coefficient given by\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370767.image6.png\" alt=\"image6.png\" width=\"48\" height=\"48\" />\r\n\r\nsubstitute the values into the formula:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370768.image7.png\" alt=\"image7.png\" width=\"232\" height=\"49\" />","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":208683,"title":"Pre-Calculus Workbook For Dummies Cheat Sheet","slug":"pre-calculus-workbook-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208683"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282497,"slug":"pre-calculus-workbook-for-dummies-3rd-edition","isbn":"9781119508809","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508800-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/pre-calculus-workbook-for-dummies-3rd-edition-cover-9781119508809-204x255.jpg","width":204,"height":255},"title":"Pre-Calculus Workbook For Dummies","testBankPinActivationLink":"https://testbanks.wiley.com","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of <i>Trigonometry For Dummies</i> and <i>Finite Math For Dummies.</i> </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b10666eb\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b1067148\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":167824},{"headers":{"creationTime":"2016-03-26T15:10:10+00:00","modifiedTime":"2021-12-21T20:39:55+00:00","timestamp":"2022-09-14T18:18:56+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"How to Identify Even and Odd Functions and their Graphs","strippedTitle":"how to identify even and odd functions and their graphs","slug":"how-to-identify-even-and-odd-functions-and-their-graphs","canonicalUrl":"","seo":{"metaDescription":"Learn the definitions of even and odd functions in calculus so you can determine which half of the points you'll need to graph.","noIndex":0,"noFollow":0},"content":"Knowing whether a function is even or odd helps you to graph it because that information tells you which half of the points you have to graph. These types of functions are symmetrical, so whatever is on one side is exactly the same as the other side.\r\n\r\nIf a function is even, the graph is symmetrical about the <i>y-</i>axis. If the function is odd, the graph is symmetrical about the origin.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Even function: </b>The mathematical definition of an <i>even function</i> is <i>f</i>(–<i>x</i>) = <i>f</i>(<i>x</i>) for any value of <i>x.</i> The simplest example of this is <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> because <i>f(x)=f(-x)</i> for all <i>x</i>. For example, <i>f</i>(3) = 9, and <i>f</i>(–3) = 9. Basically, the opposite input yields the same output.</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"466\"]<img src=\"https://www.dummies.com/wp-content/uploads/370864.image0.jpg\" alt=\"image0.jpg\" width=\"466\" height=\"400\" /> Visually speaking, the graph is a mirror image about the y-axis, as shown here.[/caption]</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Odd function: </b>The definition of an <i>odd function</i> is <i>f</i>(–<i>x</i>) = <i>–f</i>(<i>x</i>) for any value of <i>x.</i> The opposite input gives the opposite output. These graphs have 180-degree symmetry about the origin. If you turn the graph upside down, it looks the same.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370865.image1.jpg\" alt=\"image1.jpg\" width=\"400\" height=\"397\" />\r\n<p class=\"child-para\">The example shown above, <i>f</i>(<i>x</i>) = <i>x</i><sup>3</sup>, is an odd function because <i>f(-x)=-f(x)</i> for all <i>x</i>. For example, <i>f</i>(3) = 27 and <i>f</i>(–3) = –27.</p>\r\n</li>\r\n</ul>","description":"Knowing whether a function is even or odd helps you to graph it because that information tells you which half of the points you have to graph. These types of functions are symmetrical, so whatever is on one side is exactly the same as the other side.\r\n\r\nIf a function is even, the graph is symmetrical about the <i>y-</i>axis. If the function is odd, the graph is symmetrical about the origin.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Even function: </b>The mathematical definition of an <i>even function</i> is <i>f</i>(–<i>x</i>) = <i>f</i>(<i>x</i>) for any value of <i>x.</i> The simplest example of this is <i>f</i>(<i>x</i>) = <i>x</i><sup>2</sup> because <i>f(x)=f(-x)</i> for all <i>x</i>. For example, <i>f</i>(3) = 9, and <i>f</i>(–3) = 9. Basically, the opposite input yields the same output.</p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"466\"]<img src=\"https://www.dummies.com/wp-content/uploads/370864.image0.jpg\" alt=\"image0.jpg\" width=\"466\" height=\"400\" /> Visually speaking, the graph is a mirror image about the y-axis, as shown here.[/caption]</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Odd function: </b>The definition of an <i>odd function</i> is <i>f</i>(–<i>x</i>) = <i>–f</i>(<i>x</i>) for any value of <i>x.</i> The opposite input gives the opposite output. These graphs have 180-degree symmetry about the origin. If you turn the graph upside down, it looks the same.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370865.image1.jpg\" alt=\"image1.jpg\" width=\"400\" height=\"397\" />\r\n<p class=\"child-para\">The example shown above, <i>f</i>(<i>x</i>) = <i>x</i><sup>3</sup>, is an odd function because <i>f(-x)=-f(x)</i> for all <i>x</i>. For example, <i>f</i>(3) = 27 and <i>f</i>(–3) = –27.</p>\r\n</li>\r\n</ul>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":208683,"title":"Pre-Calculus Workbook For Dummies Cheat Sheet","slug":"pre-calculus-workbook-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/208683"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282497,"slug":"pre-calculus-workbook-for-dummies-3rd-edition","isbn":"9781119508809","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119508800-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119508800/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/pre-calculus-workbook-for-dummies-3rd-edition-cover-9781119508809-204x255.jpg","width":204,"height":255},"title":"Pre-Calculus Workbook For Dummies","testBankPinActivationLink":"https://testbanks.wiley.com","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of <i>Trigonometry For Dummies</i> and <i>Finite Math For Dummies.</i> </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b105dcfc\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508809&quot;]}]\" id=\"du-slot-63221b105e589\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":167765}],"_links":{"self":{"self":"https://dummies-api.dummies.com/v2/categories/33727/categoryArticles?sortField=time&sortOrder=1&size=10&offset=0"},"next":{"self":"https://dummies-api.dummies.com/v2/categories/33727/categoryArticles?sortField=time&sortOrder=1&size=10&offset=10"},"last":{"self":"https://dummies-api.dummies.com/v2/categories/33727/categoryArticles?sortField=time&sortOrder=1&size=10&offset=196"}}},"objectTitle":"","status":"success","pageType":"article-category","objectId":"33727","page":1,"sortField":"time","sortOrder":1,"categoriesIds":[],"articleTypes":[],"filterData":{"categoriesFilter":[{"itemId":0,"itemName":"All Categories","count":206}],"articleTypeFilter":[{"articleType":"All Types","count":206},{"articleType":"Articles","count":202},{"articleType":"Cheat Sheet","count":3},{"articleType":"Step by Step","count":1}]},"filterDataLoadedStatus":"success","pageSize":10},"adsState":{"pageScripts":{"headers":{"timestamp":"2022-10-18T10:50:01+00:00"},"adsId":0,"data":{"scripts":[{"pages":["all"],"location":"header","script":"<!--Optimizely Script-->\r\n<script src=\"https://cdn.optimizely.com/js/10563184655.js\"></script>","enabled":false},{"pages":["all"],"location":"header","script":"<!-- comScore Tag -->\r\n<script>var _comscore = _comscore || [];_comscore.push({ c1: \"2\", c2: \"15097263\" });(function() {var s = document.createElement(\"script\"), el = document.getElementsByTagName(\"script\")[0]; s.async = true;s.src = (document.location.protocol == \"https:\" ? \"https://sb\" : \"http://b\") + \".scorecardresearch.com/beacon.js\";el.parentNode.insertBefore(s, el);})();</script><noscript><img src=\"https://sb.scorecardresearch.com/p?c1=2&c2=15097263&cv=2.0&cj=1\" /></noscript>\r\n<!-- / comScore Tag -->","enabled":true},{"pages":["all"],"location":"footer","script":"<!--BEGIN QUALTRICS WEBSITE FEEDBACK SNIPPET-->\r\n<script type='text/javascript'>\r\n(function(){var g=function(e,h,f,g){\r\nthis.get=function(a){for(var a=a+\"=\",c=document.cookie.split(\";\"),b=0,e=c.length;b<e;b++){for(var d=c[b];\" \"==d.charAt(0);)d=d.substring(1,d.length);if(0==d.indexOf(a))return d.substring(a.length,d.length)}return null};\r\nthis.set=function(a,c){var b=\"\",b=new Date;b.setTime(b.getTime()+6048E5);b=\"; expires=\"+b.toGMTString();document.cookie=a+\"=\"+c+b+\"; path=/; \"};\r\nthis.check=function(){var a=this.get(f);if(a)a=a.split(\":\");else if(100!=e)\"v\"==h&&(e=Math.random()>=e/100?0:100),a=[h,e,0],this.set(f,a.join(\":\"));else return!0;var c=a[1];if(100==c)return!0;switch(a[0]){case \"v\":return!1;case \"r\":return c=a[2]%Math.floor(100/c),a[2]++,this.set(f,a.join(\":\")),!c}return!0};\r\nthis.go=function(){if(this.check()){var a=document.createElement(\"script\");a.type=\"text/javascript\";a.src=g;document.body&&document.body.appendChild(a)}};\r\nthis.start=function(){var t=this;\"complete\"!==document.readyState?window.addEventListener?window.addEventListener(\"load\",function(){t.go()},!1):window.attachEvent&&window.attachEvent(\"onload\",function(){t.go()}):t.go()};};\r\ntry{(new g(100,\"r\",\"QSI_S_ZN_5o5yqpvMVjgDOuN\",\"https://zn5o5yqpvmvjgdoun-wiley.siteintercept.qualtrics.com/SIE/?Q_ZID=ZN_5o5yqpvMVjgDOuN\")).start()}catch(i){}})();\r\n</script><div id='ZN_5o5yqpvMVjgDOuN'><!--DO NOT REMOVE-CONTENTS PLACED HERE--></div>\r\n<!--END WEBSITE FEEDBACK SNIPPET-->","enabled":false},{"pages":["all"],"location":"header","script":"<!-- Hotjar Tracking Code for http://www.dummies.com -->\r\n<script>\r\n (function(h,o,t,j,a,r){\r\n h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)};\r\n h._hjSettings={hjid:257151,hjsv:6};\r\n a=o.getElementsByTagName('head')[0];\r\n r=o.createElement('script');r.async=1;\r\n r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv;\r\n a.appendChild(r);\r\n })(window,document,'https://static.hotjar.com/c/hotjar-','.js?sv=');\r\n</script>","enabled":false},{"pages":["article"],"location":"header","script":"<!-- //Connect Container: dummies --> <script src=\"//get.s-onetag.com/bffe21a1-6bb8-4928-9449-7beadb468dae/tag.min.js\" async defer></script>","enabled":true},{"pages":["homepage"],"location":"header","script":"<meta name=\"facebook-domain-verification\" content=\"irk8y0irxf718trg3uwwuexg6xpva0\" />","enabled":true},{"pages":["homepage","article","category","search"],"location":"footer","script":"<!-- Facebook Pixel Code -->\r\n<noscript>\r\n<img height=\"1\" width=\"1\" src=\"https://www.facebook.com/tr?id=256338321977984&ev=PageView&noscript=1\"/>\r\n</noscript>\r\n<!-- End Facebook Pixel Code -->","enabled":true}]}},"pageScriptsLoadedStatus":"success"},"navigationState":{"navigationCollections":[{"collectionId":287568,"title":"BYOB (Be Your Own Boss)","hasSubCategories":false,"url":"/collection/for-the-entry-level-entrepreneur-287568"},{"collectionId":293237,"title":"Be a Rad Dad","hasSubCategories":false,"url":"/collection/be-the-best-dad-293237"},{"collectionId":294090,"title":"Contemplating the Cosmos","hasSubCategories":false,"url":"/collection/theres-something-about-space-294090"},{"collectionId":287563,"title":"For Those Seeking Peace of Mind","hasSubCategories":false,"url":"/collection/for-those-seeking-peace-of-mind-287563"},{"collectionId":287570,"title":"For the Aspiring Aficionado","hasSubCategories":false,"url":"/collection/for-the-bougielicious-287570"},{"collectionId":291903,"title":"For the Budding Cannabis Enthusiast","hasSubCategories":false,"url":"/collection/for-the-budding-cannabis-enthusiast-291903"},{"collectionId":291934,"title":"For the Exam-Season Crammer","hasSubCategories":false,"url":"/collection/for-the-exam-season-crammer-291934"},{"collectionId":287569,"title":"For the Hopeless Romantic","hasSubCategories":false,"url":"/collection/for-the-hopeless-romantic-287569"},{"collectionId":287567,"title":"For the Unabashed Hippie","hasSubCategories":false,"url":"/collection/for-the-unabashed-hippie-287567"},{"collectionId":292186,"title":"Just DIY It","hasSubCategories":false,"url":"/collection/just-diy-it-292186"}],"navigationCollectionsLoadedStatus":"success","navigationCategories":{"books":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/books/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/books/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/books/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/books/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/books/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/books/level-0-category-0"}},"articles":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/articles/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/articles/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/articles/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/articles/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/articles/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/articles/level-0-category-0"}}},"navigationCategoriesLoadedStatus":"success"},"searchState":{"searchList":[],"searchStatus":"initial","relatedArticlesList":[],"relatedArticlesStatus":"initial"},"routeState":{"name":"ArticleCategory","path":"/category/articles/pre-calculus-33727/","hash":"","query":{},"params":{"category":"pre-calculus-33727"},"fullPath":"/category/articles/pre-calculus-33727/","meta":{"routeType":"category","breadcrumbInfo":{"suffix":"Articles","baseRoute":"/category/articles"},"prerenderWithAsyncData":true},"from":{"name":null,"path":"/","hash":"","query":{},"params":{},"fullPath":"/","meta":{}}},"sfmcState":{"status":"initial"},"profileState":{"auth":{},"userOptions":{},"status":"success"}}
Logo
  • Articles Open Article Categories
  • Books Open Book Categories
  • Collections Open Collections list
  • Custom Solutions

Article Categories

Book Categories

Collections

Explore all collections
BYOB (Be Your Own Boss)
Be a Rad Dad
Contemplating the Cosmos
For Those Seeking Peace of Mind
For the Aspiring Aficionado
For the Budding Cannabis Enthusiast
For the Exam-Season Crammer
For the Hopeless Romantic
For the Unabashed Hippie
Just DIY It
Log In
  • Home
  • Academics & The Arts Articles
  • Math Articles
  • Pre-Calculus Articles

Pre-Calculus Articles

Can learning (or reviewing) pre-calc really be fun? With our breakdowns and practice problems, it'll come pretty close.

Articles From Pre-Calculus

page 1
page 2
page 3
page 4
page 5
page 6
page 7
page 8
page 9
page 10
page 11
page 12
page 13
page 14
page 15
page 16
page 17
page 18
page 19
page 20
page 21

Filter Results

206 results
206 results
Pre-Calculus Pre-Calculus: Test the Roots By Long Division of Polynomials

Article / Updated 10-06-2022

Once you have used the rational root theorem to list all the possible rational roots of any polynomial, the next step is to test the roots. One way is to use long division of polynomials and hope that when you divide you get a remainder of 0. Once you have a list of possible rational roots, you then pick one and assume that it’s a root. For example, consider the equation f(x) = 2x4 – 9x3 – 21x2 + 88x + 48, which has the following possible rational roots: If x = c is a root, then x – c is a factor. So if you pick x = 2 as your guess for the root, x – 2 should be a factor. You can use long division to test if x – 2 is actually a factor and, therefore, x = 2 is a root. Dividing polynomials to get a specific answer isn’t something you do every day, but the idea of a function or expression that’s written as the quotient of two polynomials is important for pre-calculus. If you divide a polynomial by another and get a remainder of 0, the divisor is a factor, which in turn gives a root. In math lingo, the division algorithm states the following: If f(x) and d(x) are polynomials such that d(x) isn’t equal to 0, and the degree of d(x) isn’t larger than the degree of f(x), there are unique polynomials q(x) and r(x) such that In plain English, the dividend equals the divisor times the quotient plus the remainder. You can always check your results by remembering this information. Remember the mnemonic device Dirty Monkeys Smell Bad when doing long division to check your roots. Make sure all terms in the polynomial are listed in descending order and that every degree is represented. In other words, if x2 is missing, put in a placeholder of 0x2 and then do the division. (This step is just to make the division process easier.) To divide two polynomials, follow these steps: Divide. Divide the leading term of the dividend by the leading term of the divisor. Write this quotient directly above the term you just divided into. Multiply. Multiply the quotient term from Step 1 by the entire divisor. Write this polynomial under the dividend so that like terms are lined up. Subtract. Subtract the whole line you just wrote from the dividend. You can change all the signs and add if it makes you feel more comfortable. This way, you won’t forget signs. Bring down the next term. Do exactly what this says; bring down the next term in the dividend. Repeat Steps 1–4 over and over until the remainder polynomial has a degree that’s less than the dividend’s. The following list explains how to divide 2x4 – 9x3 – 21x2 + 88x + 48 by x – 2. Each step corresponds with the numbered step in the illustration in this figure. The process of long division of polynomials. (Note that using Descartes’s rule of signs, you find that this particular example may have positive roots, so it’s efficient to try a positive number here. If Descartes’s rule of signs had said that no positive roots existed, you wouldn’t test any positives!) Divide. What do you have to multiply x in the divisor by to make it become 2x4 in the dividend? The quotient, 2x3, goes above the 2x4 term. Multiply. Multiply this quotient by the divisor and write it under the dividend. Subtract. Subtract this line from the dividend: (2x4 – 9x3) – (2x4 – 4x3) = –5x3. If you’ve done the job right, the subtraction of the first terms always produces 0. Bring down. Bring down the other terms of the dividend. Divide. What do you have to multiply x by to make it –5x3? Put the answer, –5x2, above the –21x2. Multiply. Multiply the –5x2 times the x – 2 to get –5x3 + 10x2. Write it under the remainder with the degrees lined up. Subtract. You now have (–5x3 – 21x2) – (–5x3 + 10x2) = –31x2. Bring down. The +88x takes its place. Divide. What to multiply by to make x become –31x2? The quotient –31x goes above –21x2. Multiply. The value –31x times (x – 2) is –31x2 + 62x; write it under the remainder. Subtract. You now have (–31x2 + 88x) – (–31x2 + 62x), which is 26x. Bring down. The +48 comes down. Divide. The term 26x divided by x is 26. This answer goes on top. Multiply. The constant 26 multiplied by (x – 2) is 26x – 52. Subtract. You subtract (26x + 48) – (26x – 52) to get 100. Stop. The remainder 100 has a degree that’s less than the divisor of x – 2. Wow . . . now you know why they call it long division. You went through all that to find out that x – 2 isn’t a factor of the polynomial, which means that x = 2 isn’t a root. If you divide by c and the remainder is 0, then the linear expression (x – c) is a factor and that c is a root. A remainder other than 0 implies that (x – c) isn’t a factor and that c isn’t a root.

View Article
Pre-Calculus Pre-Calculus: Finding the General Formula for the nth Term

Article / Updated 10-06-2022

At some point, your pre-calculus teacher will ask you to find the general formula for the nth term of an arithmetic sequence without knowing the first term or the common difference. In this case, you will be given two terms (not necessarily consecutive), and you will use this information to find a1 and d. The steps are: Find the common difference d, write the specific formula for the given sequence, and then find the term you're looking for. For instance, to find the general formula of an arithmetic sequence where a4 = –23 and a22 = 40, follow these steps: Find the common difference. You have to be creative in finding the common difference for these types of problems. a.Use the formula an = a1 + (n – 1)d to set up two equations that use the given information. For the first equation, you know that when n = 4, an = –23: –23 = a1 + (4 – 1)d –23 = a1 + 3d For the second equation, you know that when n = 22, an = 40: 40 = a1 + (22 – 1)d 40 = a1 + 21d b.Set up a system of equations and solve for d. The system looks like this: You can use elimination or substitution to solve the system. Elimination works nicely because you can multiply either equation by –1 and add the two together to get 63 = 18d. Therefore, d = 3.5. Write the formula for the specific sequence. This step involves a little work. a.Plug d into one of the equations to solve for a1. You can plug 3.5 back into either equation: –23 = a1 + 3(3.5), or a1 = –33.5. b.Use a1 and d to find the general formula for an. This step becomes a simple three-step simplification: an = –33.5 + (n – 1)3.5 an = –33.5 + 3.5n – 3.5 an = 3.5n – 37 Find the term you were looking for. In this example, you weren't asked to find any specific term (always read the directions!), but if you were, you could plug that number in for n and then find the term you were looking for.

View Article
Pre-Calculus Finding the Cosine of Sums and Differences of Angles

Article / Updated 09-22-2022

You can use the sum and difference formulas for cosine to calculate the cosine of the sums and differences of angles similarly to the way you can use the sum and difference formulas for sine, because the formulas look very similar to each other. When working with sines and cosines of sums and differences of angles, you're simply plugging in given values for the variables (angles). Just make sure you use the correct formula based on the information you're given in the question. Here are the sum and difference formulas for cosines: The sum and difference formulas for cosine (and sine) can do more than calculate a trig value for an angle not marked on the unit circle (at least for angles that are multiples of 15 degrees). They can also be used to find the cosine (and sine) of the sum or difference of two angles based on information given about the two angles. For such problems, you'll be given two angles (call them A and B), the sine or cosine of A and B, and the quadrant(s) in which the two angles are located. Use the following steps to find the exact value of cos(A + B), given that cos A = –3/5, with A in quadrant II of the coordinate plane, and sin B = –7/25, with B in quadrant III: Choose the appropriate formula and substitute the information you know to determine the missing information. then substitutions result in this equation: To proceed any further, you need to find cos B and sin A. Draw pictures representing right triangles in the quadrant(s). Drawing pictures helps you visualize the missing pieces of info. You need to draw one triangle for angle A in quadrant II and one for angle B in quadrant III. Using the definition of sine as opp/hyp and cosine as adj/hyp, this figure shows these triangles. Notice that the value of a leg is missing in each triangle. To find the missing values, use the Pythagorean theorem. The length of the missing leg in Figure a is 4, and the length of the missing leg in Figure b is –24. Determine the missing trig ratios to use in the sum or difference formula. You use the definition of cosine to find that cos B = –24/25 and the definition of sine to find that sin A = 4/5. Substitute the missing trig ratios into the sum or difference formula and simplify. You now have this equation: Follow the order of operations to get this answer: This equation simplifies to cos(A + B) = 4/5.

View Article
Pre-Calculus A Quick Guide to the 30-60-90 Triangle

Article / Updated 08-08-2022

The 30-60-90 triangle is shaped like half of an equilateral triangle, cut straight down the middle along its altitude. It has angles of 30 degrees, 60 degrees, and 90 degrees, thus, its name! In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3. The hypotenuse is the longest side in a right triangle, which is different from the long leg. The long leg is the leg opposite the 60-degree angle. Two of the most common right triangles are 30-60-90 and the 45-45-90-degree triangles. All 30-60-90 triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in radians, it translates to the following: The figure illustrates the ratio of the sides for the 30-60-90-degree triangle. If you know one side of a 30-60-90 triangle, you can find the other two by using shortcuts. Here are the three situations you come across when doing these calculations: Type 1: You know the short leg (the side across from the 30-degree angle). Double its length to find the hypotenuse. You can multiply the short side by the square root of 3 to find the long leg. Type 2: You know the hypotenuse. Divide the hypotenuse by 2 to find the short side. Multiply this answer by the square root of 3 to find the long leg. Type 3: You know the long leg (the side across from the 60-degree angle). Divide this side by the square root of 3 to find the short side. Double that figure to find the hypotenuse. In the triangle TRI in this figure, the hypotenuse is 14 inches long; how long are the other sides? Because you have the hypotenuse TR = 14, you can divide by 2 to get the short side: RI = 7. Now you multiply this length by the square root of 3 to get the long side:

View Article
Pre-Calculus Pre-Calculus Workbook For Dummies Cheat Sheet

Cheat Sheet / Updated 04-06-2022

Pre-calculus uses the information you know from Algebra I and II and ratchets up the difficulty level to prepare you for calculus. This cheat sheet is designed to help you review key formulas and functions on the fly as you study. It includes formulas, the laws of logarithmic functions, trigonometric values of basic angles, conic section equations, and interval notation.

View Cheat Sheet
Pre-Calculus Pre-Calculus: 1001 Practice Problems For Dummies Cheat Sheet

Cheat Sheet / Updated 02-24-2022

Pre-calculus draws from algebra, geometry, and trigonometry and combines these topics to prepare you for the techniques you need to succeed in calculus. This cheat sheet provides the most frequently used formulas, with brief descriptions of what the letters and symbols represent. Counting techniques are also here, letting you count numbers of events without actually having to list all the ways to do them. Also, you find a step-by-step description of how to complete the square — most useful when you’re working with conic sections and other equations with specific formats.

View Cheat Sheet
Pre-Calculus How to Find Imaginary Roots Using the Fundamental Theorem of Algebra

Article / Updated 12-21-2021

The fundamental theorem of algebra can help you find imaginary roots. Imaginary roots appear in a quadratic equation when the discriminant of the quadratic equation — the part under the square root sign (b2 – 4ac) — is negative. If this value is negative, you can’t actually take the square root, and the answers are not real. In other words, there is no real solution; therefore, the graph won’t cross the x-axis. Using the quadratic formula always gives you two solutions, because the plus/minus sign means you’re both adding and subtracting and getting two completely different answers. When the number underneath the square-root sign in the quadratic formula is negative, the answers are called complex conjugates. One is r + si and the other is r – si. These numbers have both real (the r) and imaginary (the si) parts. The complex number system consists of all numbers r + si where r and s are real numbers. Observe that when s = 0, you simply have the real numbers. Therefore the real numbers are a subset of the complex number system. The fundamental theorem of algebra says that every polynomial function has at least one root in the complex number system. The highest degree of a polynomial gives you the highest possible number of distinct complex roots for the polynomial. Between this fact and Descartes’s rule of signs, you can get an idea of how many imaginary roots a polynomial has. Here’s how Descartes’s rule of signs can give you the numbers of possible real roots, both positive and negative: Positive real roots. For the number of positive real roots, look at the polynomial, written in descending order, and count how many times the sign changes from term to term. This value represents the maximum number of positive roots in the polynomial. For example, in the polynomial f(x) = 2x4 – 9x3 – 21x2 + 88x + 48, you see two changes in sign (don’t forget to include the sign of the first term!) — from the first term (+2x4) to the second (-9x3) and from the third term (-21x2) to the fourth term (88x). That means this equation can have up to two positive solutions. Descartes’s rule of signs says the number of positive roots is equal to changes in sign of f(x), or is less than that by an even number (so you keep subtracting 2 until you get either 1 or 0). Therefore, the previous f(x) may have 2 or 0 positive roots. Negative real roots. For the number of negative real roots, find f(–x) and count again. Because negative numbers raised to even powers are positive and negative numbers raised to odd powers are negative, this change affects only terms with odd powers. This step is the same as changing each term with an odd degree to its opposite sign and counting the sign changes again, which gives you the maximum number of negative roots. The example equation becomes f(–x) = 2x4 + 9x3 – 21x2 – 88x + 48, which changes signs twice. There can be, at most, two negative roots. However, similar to the rule for positive roots, the number of negative roots is equal to the changes in sign for f(–x), or must be less than that by an even number. Therefore, this example can have either 2 or 0 negative roots. Pair up every possible number of positive real roots with every possible number of negative real roots; the remaining number of roots for each situation represents the number of imaginary roots. For example, the polynomial f(x) = 2x4 – 9x3 – 21x2 + 88x + 48 has a degree of 4, with two or zero positive real roots, and two or zero negative real roots. With this information, you can pair up the possible situations: Two positive and two negative real roots, with zero imaginary roots Two positive and zero negative real roots, with two imaginary roots Zero positive and two negative real roots, with two imaginary roots Zero positive and zero negative real roots, with four imaginary roots The following chart makes the information easier to picture: Positive real roots Negative real roots Imaginary roots 2 2 0 2 0 2 0 2 2 0 0 4 Complex numbers are written in the form r + si and have both a real and an imaginary part, which is why every polynomial has at least one root in the complex number system. Real and imaginary numbers are both included in the complex number system. Real numbers have no imaginary part, and pure imaginary numbers have no real part. For example, if x = 7 is one root of the polynomial, this root is considered both real and complex because it can be rewritten as x = 7 + 0i (the imaginary part is 0). The fundamental theorem of algebra gives the total number of complex roots (say there are seven); Descartes’s rule of signs tells you how many possible real roots exist and how many of them are positive and negative (say there are, at most, two positive roots but only one negative root). Now, assume you’ve found them all: x = 1, x = 7, and x = –2. These roots are real, but they’re also complex because they can all be rewritten. The first two columns in the chart find the real roots and classify them as positive or negative. The third column is actually finding, specifically, the non-real numbers: complex numbers with non-zero imaginary parts.

View Article
Pre-Calculus How to Graph Polar Coordinates with Negative Values

Article / Updated 12-21-2021

Sometimes your geometry teacher may spice things up a bit with complicated polar coordinates — points with negative angles and/or radii. The following list shows you how to plot in three situations — when the angle is negative, when the radius is negative, and when both are negative. When the angle is negative: Negative angles move in a clockwise direction. This figure shows an example point, D. To locate the polar coordinate point D at first locate the angle and then find the location of the radius, 1, on that line. When the radius is negative: When graphing a polar coordinate with a negative radius, you move from the pole in the direction opposite the given positive angle (on the same line as the given angle but in the direction opposite to the angle from the pole). For example, check out point F at in the figure. Some teachers prefer to teach their students to move right along the x- (polar) axis for positive numbers (radii) and left for negative. Then you do the rotation for the angle in a positive direction. You’ll get to the same spot with that method. For example, take a look at point F in the figure. Because the radius is negative, move along the left x-axis 1/2 of a unit. Then rotate the angle in the positive direction (counterclockwise) pi/3 radians. You should arrive at your destination, point F. When both the angle and radius are negative: To express a polar coordinate with a negative radius and a negative angle, locate the terminal side of the negative angle first and then move in the opposite direction to locate the radius. For example, point G in the figure has these characteristics at Indeed, except the origin, each given point can have the following four types of representations: Positive radius, positive angle Positive radius, negative angle Negative radius, positive angle Negative radius, negative angle For example, point E in the figure can have three other polar coordinate representations with different combinations of signs for the radius and angle: When polar graphing, you can change the coordinate of any point you’re given into polar coordinates that are easy to deal with (such as positive radius, positive angle).

View Article
Pre-Calculus How to Find Binomial Coefficients

Article / Updated 12-21-2021

Depending on how many times you must multiply the same binomial — a value also known as an exponent — the binomial coefficients for that particular exponent are always the same. The binomial coefficients are found by using the combinations formula. If the exponent is relatively small, you can use a shortcut called Pascal's triangle to find these coefficients. If not, you can always rely on algebra! Pascal's triangle, named after the famous mathematician Blaise Pascal, names the binomial coefficients for the binomial expansion. It is especially useful when raising a binomial to lower degrees. For example, if a sadistic teacher asked you to find (3x + 4)10, you probably wouldn't want to use Pascal's triangle; instead, you'd just use the algebraic formula described shortly. The figure illustrates this concept. The top number of the triangle is 1, as well as all the numbers on the outer sides. To get any term in the triangle, you find the sum of the two numbers above it. Each row gives the coefficients to (a + b)n, starting with n = 0. To find the binomial coefficients for (a + b)n, use the nth row and always start with the beginning. For instance, the binomial coefficients for (a + b)5 are 1, 5, 10, 10, 5, and 1 — in that order. If you need to find the coefficients of binomials algebraically, there is a formula for that as well. The rth coefficient for the nth binomial expansion is written in the following form: You may recall the term factorial from your earlier math classes. If not, here is a reminder: n!, which reads as "n factorial," is defined as You read the expression for the binomial coefficient as "n choose r." You usually can find a button for combinations on a calculator. If not, you can use the factorial button and do each part separately. To make things a little easier, 0! is defined as 1. Therefore, you have these equalities: For example, to find the binomial coefficient given by substitute the values into the formula:

View Article
Pre-Calculus How to Identify Even and Odd Functions and their Graphs

Article / Updated 12-21-2021

Knowing whether a function is even or odd helps you to graph it because that information tells you which half of the points you have to graph. These types of functions are symmetrical, so whatever is on one side is exactly the same as the other side. If a function is even, the graph is symmetrical about the y-axis. If the function is odd, the graph is symmetrical about the origin. Even function: The mathematical definition of an even function is f(–x) = f(x) for any value of x. The simplest example of this is f(x) = x2 because f(x)=f(-x) for all x. For example, f(3) = 9, and f(–3) = 9. Basically, the opposite input yields the same output. Odd function: The definition of an odd function is f(–x) = –f(x) for any value of x. The opposite input gives the opposite output. These graphs have 180-degree symmetry about the origin. If you turn the graph upside down, it looks the same. The example shown above, f(x) = x3, is an odd function because f(-x)=-f(x) for all x. For example, f(3) = 27 and f(–3) = –27.

View Article
page 1
page 2
page 3
page 4
page 5
page 6
page 7
page 8
page 9
page 10
page 11
page 12
page 13
page 14
page 15
page 16
page 17
page 18
page 19
page 20
page 21

Quick Links

  • About For Dummies
  • Contact Us
  • Activate A Book Pin

Connect

Opt in to our newsletter!

By entering your email address and clicking the “Submit” button, you agree to the Terms of Use and Privacy Policy & to receive electronic communications from Dummies.com, which may include marketing promotions, news and updates.

About Dummies

Dummies has always stood for taking on complex concepts and making them easy to understand. Dummies helps everyone be more knowledgeable and confident in applying what they know. Whether it's to pass that big test, qualify for that big promotion or even master that cooking technique; people who rely on dummies, rely on it to learn the critical skills and relevant information necessary for success.

Terms of Use
Privacy Policy
Cookies Settings
Do Not Sell My Personal Info - CA Only