{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2022-10-18T16:01:26+00:00"},"categoryId":33721,"data":{"title":"Algebra","slug":"algebra","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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"parentCategory":{"categoryId":33720,"title":"Math","slug":"math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"}},"childCategories":[],"description":"Master algebra and algebra II with quick breakdowns of key math skills, plus practice problem sets and other cool stuff.","relatedArticles":{"self":"https://dummies-api.dummies.com/v2/articles?category=33721&offset=0&size=5"},"hasArticle":true,"hasBook":true,"articleCount":207,"bookCount":13},"_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":206,"items":[{"headers":{"creationTime":"2022-10-04T19:48:15+00:00","modifiedTime":"2022-10-10T12:23:08+00:00","timestamp":"2022-10-10T15:01:02+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II All-in-One For Dummies Cheat Sheet","strippedTitle":"algebra ii all-in-one for dummies cheat sheet","slug":"algebra-ii-all-in-one-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Here it is. You have this All-in-One reference for concepts and formulas occurring in Algebra II. The material here is grouped by general algebraic content to m","noIndex":0,"noFollow":0},"content":"Here it is. You have this All-in-One reference for concepts and formulas occurring in Algebra II. The material here is grouped by general algebraic content to make it easier to find what you need. The formulas have the standard mathematical format with variables appearing as <em>x</em>, <em>y</em>, and <em>z</em> and the constant numbers appearing as letters at the beginning of the alphabet.","description":"Here it is. You have this All-in-One reference for concepts and formulas occurring in Algebra II. The material here is grouped by general algebraic content to make it easier to find what you need. The formulas have the standard mathematical format with variables appearing as <em>x</em>, <em>y</em>, and <em>z</em> and the constant numbers appearing as letters at the beginning of the alphabet.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" \t <p><b>Mary Jane Sterling</b> is the author of numerous <i>For Dummies</i> books. She has been teaching at Bradley University in Peoria, Illinois, for more than 25 years. 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","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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119843047&quot;]}]\" id=\"du-slot-634433ae99ecf\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119843047&quot;]}]\" id=\"du-slot-634433ae9a832\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":0,"title":"","slug":null,"categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/"}}],"content":[{"title":"Line formulas","thumb":null,"image":null,"content":"<p>When graphing segments and lines containing points on the coordinate plane, you have the opportunity to find many values: slope, distance, midpoint, and so on. When using these formulas, you see the coordinates of points written in (<em>x</em><sub>a</sub>, <em>y</em><sub>a</sub>) format. Note that the slope of a line is designated with a small <em>m</em> and the midpoint of a segment with a capital <em>M</em>.</p>\n<p>Formulas to use when given the points (<em>x</em><sub>1</sub>,<em>y</em><sub>1</sub>) and (<em>x</em><sub>2</sub>,<em>y</em><sub>2</sub>).</p>\n<p>Slope of line through the points:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295155\" src=\"https://www.dummies.com/wp-content/uploads/slope-of-line-through-points-formula.png\" alt=\"\" width=\"91\" height=\"52\" /></p>\n<p>Midpoint of the segment between the points:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295157\" src=\"https://www.dummies.com/wp-content/uploads/midpoint-of-segment-between-points-formula.png\" alt=\"\" width=\"179\" height=\"55\" /></p>\n<p>Distance between the points:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295156\" src=\"https://www.dummies.com/wp-content/uploads/distance-between-points-formula.png\" alt=\"\" width=\"211\" height=\"39\" /></p>\n<p>Given the equation in slope-intercept form: <em>y = mx + b</em></p>\n<p>Slope of a parallel line: <em>m</em></p>\n<p>Slope of a perpendicular line:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295160\" src=\"https://www.dummies.com/wp-content/uploads/slope-of-perpendicular-line.png\" alt=\"\" width=\"42\" height=\"49\" /></p>\n<p>Given the standard form of a line: <em>Ax + By = C</em></p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295158\" src=\"https://www.dummies.com/wp-content/uploads/x-intercept-y-intercept-slope.png\" alt=\"x-intercept: (C÷A,0) ; y-intercept: (C÷B,0) ; slope: -A÷B\" width=\"431\" height=\"60\" /></p>\n"},{"title":"Miscellaneous formulas","thumb":null,"image":null,"content":"<p>When you can’t find a particular formula under any of the other headings, this is where you look. There’s a little geometry, some counting rules, and formats for basic factoring.</p>\n<p><strong>Distance</strong>:</p>\n<p><em>d = rt</em>, where <em>r</em> is the rate of speed and <em>t</em> is the time in that same rate.</p>\n<p><strong>Quadratic formula:</strong></p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295162\" src=\"https://www.dummies.com/wp-content/uploads/quadratic-formula.png\" alt=\"When ax2 + bx + c = 0, x = -b ± √b2 - 4ac ÷ 2a\" width=\"346\" height=\"58\" /></p>\n<p><strong>Factorial</strong>:</p>\n<p><em>n</em>! = <em>n</em> ∙ (<em>n</em> – 1) ∙ (<em>n</em> – 2) ∙ (<em>n</em> – 3) ∙∙∙ 3 ∙ 2 ∙ 1, where <em>n</em> is a non-negative integer.</p>\n<blockquote><p>Special rule: 0! = 1</p></blockquote>\n<p>Absolute value:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295161\" src=\"https://www.dummies.com/wp-content/uploads/absolute-value-formula.png\" alt=\"| a | = { a if a ≥ 0 -a if a &lt; 0\" width=\"139\" height=\"57\" /></p>\n<p><strong>Factoring</strong>:</p>\n<blockquote><p>Difference of squares:</p>\n<p><em>a</em>² – <em>b</em>² = (<em>a</em> – <em>b</em>)(<em>a</em> + <em>b</em>)</p>\n<p>Difference of cubes:</p>\n<p><em>a</em>³ – <em>b</em>³ = (<em>a</em> – <em>b</em>)(<em>a</em>² + <em>ab</em> + <em>b</em>²)</p>\n<p>Sum of cubes:</p>\n<p><em>a</em>³ + <em>b</em>³ = (<em>a</em> + <em>b</em>)(<em>a</em>² &#8211; <em>ab</em> + <em>b</em>²)</p></blockquote>\n<p><strong>Counting</strong>:</p>\n<blockquote><p>Multiplication property: <em>m</em><sub>1</sub> ∙ <em>m</em><sub>2</sub> ∙ <em>m</em><sub>3</sub> ∙∙∙, where event 1 can happen <em>m</em><sub>1</sub> ways, event 2 can happen <em>m</em><sub>2</sub> ways, and so on.</p>\n<p>Permutations:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295164\" src=\"https://www.dummies.com/wp-content/uploads/permutations-formula.png\" alt=\"nPr = n! ÷ (n-r)!\" width=\"102\" height=\"55\" /> , where <em>n</em> is the total number of ways an event can happen and <em>r</em> is the selected number.</p>\n<p>Combinations:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295163\" src=\"https://www.dummies.com/wp-content/uploads/combinations-formula.png\" alt=\"nCr = n! ÷ (n-r)!r!\" width=\"124\" height=\"51\" /> , where <em>n</em> is the total number of ways an event can happen and <em>r</em> is the selected number.</p></blockquote>\n<p><strong>Binomial Theorem</strong>:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295165\" src=\"https://www.dummies.com/wp-content/uploads/binomial-theorem.png\" alt=\"(a+b)^n=\\sum_{k=0}^{n} \\begin{pmatrix} n\\\\ k \\end{pmatrix} a^{n-k} b^k\" width=\"201\" height=\"57\" /></p>\n<p><strong>Heron&#8217;s formula</strong>:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295166\" src=\"https://www.dummies.com/wp-content/uploads/herons-formula.png\" alt=\"A = √s(s - a)(s - b)(s - c)\" width=\"208\" height=\"32\" /></p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n"},{"title":"Inequality equivalences","thumb":null,"image":null,"content":"<p>When solving absolute value inequalities, you need to change the format so you can perform the usual algebraic processes and solve for the value of the variable.</p>\n<p>(This applies to &lt; and &gt; also):</p>\n<p>For | <em>ax</em> + <em>b</em> | ≤ <em>c</em>, solve &#8211;<em>c</em> ≤ <em>ax</em> + <em>b</em> ≤ <em>c .</em></p>\n<p>For | <em>ax</em> + <em>b</em> | ≥ <em>c</em>, solve <em>ax</em> + <em>b</em> ≤ &#8211;<em>c</em> and <em>ax</em> + <em>b</em> ≥ <em>c .</em></p>\n"},{"title":"Proportions","thumb":null,"image":null,"content":"<p>A proportion is an equation involving two ratios. Changing the initial format of a proportion can be most helpful.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295167\" src=\"https://www.dummies.com/wp-content/uploads/proportion-equation-two-ratios.png\" alt=\"a/b = c/d ↔ ad = bc ↔ b/a = d/c\" width=\"296\" height=\"52\" /></p>\n"},{"title":"Conics standard equations","thumb":null,"image":null,"content":"<p>In the case of the four basic conic sections, you have the consistent property that the center or vertex of the conic is (<em>h,k</em>).</p>\n<p><strong>Parabola</strong>:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295169\" src=\"https://www.dummies.com/wp-content/uploads/parabola-opens-upward-or-downward.png\" alt=\"y-k=a(x-h)², opens upward or downward;\" width=\"402\" height=\"35\" /> <img loading=\"lazy\" class=\"alignnone size-full wp-image-295168\" src=\"https://www.dummies.com/wp-content/uploads/parabola-opens-right-or-left.png\" alt=\"x=a(y-k)² + h, opens right or left.\" width=\"310\" height=\"29\" /></p>\n<p><strong>Circle</strong>:</p>\n<p>(<em>x</em> – <em>h</em>)<sup>2</sup> + (<em>y</em> – <em>k</em>)<sup>2</sup> = <em>r</em><sup>2</sup> , where <em>r</em> is the radius.</p>\n<p>Ellipse:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295171\" src=\"https://www.dummies.com/wp-content/uploads/ellipse-equation.png\" alt=\"(x-h)²/a² + (y-k)²/b² = 1\" width=\"179\" height=\"57\" /></p>\n<p>Hyperbola:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295172\" src=\"https://www.dummies.com/wp-content/uploads/hyperbola-opens-left-and-right-formula.png\" alt=\"(x-h)²/a² = (y-k)²/b² = 1, opens left and right.\" width=\"370\" height=\"63\" /></p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295170\" src=\"https://www.dummies.com/wp-content/uploads/hyperbola-opens-upward-and-downward-formula.png\" alt=\"(y-k)²/b² - (x-h)²/a² = 1, opens upward and downward.\" width=\"464\" height=\"57\" /></p>\n"},{"title":"Exponential and logarithmic equations and functions","thumb":null,"image":null,"content":"<p>Exponential and logarithmic functions have some special properties that allow you to simplify expressions and move from one format to another.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295173\" src=\"https://www.dummies.com/wp-content/uploads/exponential-logarithmic-equations-and-functions.png\" alt=\"\" width=\"411\" height=\"542\" /></p>\n"},{"title":"Cramer's Rule","thumb":null,"image":null,"content":"<p>Systems of linear equations can be solved using the standard algebraic processes. But Cramer’s Rule is a wonderful option when the solutions involve large and un-factorable coefficients resulting in complicated fractions.</p>\n<p>The solution of the system of linear equations:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295174\" src=\"https://www.dummies.com/wp-content/uploads/cramers-rule.png\" alt=\"Cramer's Rule formula\" width=\"426\" height=\"63\" /></p>\n"},{"title":"Properties","thumb":null,"image":null,"content":"<p>The basic properties of algebraic expressions allow you to make adjustments without changing the value of the expression. An alternate format is often more desirable.</p>\n<table>\n<tbody>\n<tr>\n<td><strong>Property</strong></td>\n<td><strong>Math Statement</strong></td>\n</tr>\n<tr>\n<td>Commutative (of addition)</td>\n<td><em>a + b = b + a</em></td>\n</tr>\n<tr>\n<td>Commutative (of multiplication)</td>\n<td><em>a ∙ b = b ∙ a</em></td>\n</tr>\n<tr>\n<td>Associative (of addition)</td>\n<td><em>a + </em>(<em>b+c</em>)<em> = </em>(<em>a+b</em>)<em> + c</em></td>\n</tr>\n<tr>\n<td>Associative (of multiplication)</td>\n<td><em>a </em>(<em>b ∙ c</em>)<em> = </em>(<em>a ∙ b</em>)<em> c</em></td>\n</tr>\n<tr>\n<td>Distributive (multiplication over addition)</td>\n<td><em>a </em>(<em>b+c</em>)<em> = a ∙ b + a ∙ c</em></td>\n</tr>\n<tr>\n<td>Distributive (multiplication over subtraction)</td>\n<td><em>a </em>(<em>b–c</em>)<em> = a ∙ b – a ∙ c</em></td>\n</tr>\n<tr>\n<td>Identity (of addition)</td>\n<td><em>a + </em>0<em> = </em>0<em> + a = a</em></td>\n</tr>\n<tr>\n<td>Identity (of multiplication)</td>\n<td><em>a</em> ∙ 1 = 1 ∙ <em>a</em> = <em>a</em></td>\n</tr>\n<tr>\n<td>Multiplication property of zero</td>\n<td><em>a ∙ b ∙ c ∙ d ∙ e ∙ f</em> = 0 → <em>a, b, c, d, e</em> or <em>f</em> = 0</td>\n</tr>\n<tr>\n<td>Additive inverse</td>\n<td><em>a</em> + (–<em>a</em>) = 0</td>\n</tr>\n<tr>\n<td>Multiplicative inverse</td>\n<td><em>a</em> ∙ (1/<em>a</em>) = 1, <em>a</em> ≠ 0</td>\n</tr>\n</tbody>\n</table>\n"},{"title":"Pascal's Triangle","thumb":null,"image":null,"content":"<p>When raising the binomial (a + b)<sup>n</sup> to a desired power, you can quickly perform the expansion using Pascal’s Triangle to help create the coefficient of each term in the expansion.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295175\" src=\"https://www.dummies.com/wp-content/uploads/pascals-triangle.png\" alt=\"Pascal's Triangle\" width=\"461\" height=\"230\" /></p>\n"},{"title":"Function transformations in graphing","thumb":null,"image":null,"content":"<p>The basic graphs of lines, parabolas, trig functions, and so on are transformed with stretches, flattening, reflections, and translations around the grid. The formulas here represent the transformation of a function ƒ(<em>x</em>) using the constants <em>h</em> and <em>a.</em></p>\n<p><strong>Translations</strong>:</p>\n<blockquote><p>Translating up: ƒ(<em>x</em>) + <em>h</em></p>\n<p>Translating down: ƒ(<em>x</em>) –<em> h</em></p>\n<p>Translating right: ƒ(<em>x</em> – <em>h</em>)</p>\n<p>Translating left: ƒ(<em>x</em> + <em>h</em>)</p></blockquote>\n<p><strong>Reflections</strong>:</p>\n<blockquote><p>Reflecting over the <em>x</em>-axis: –ƒ(<em>x</em>)</p>\n<p>Reflecting over the <em>y</em>-axis: ƒ(–<em>x</em>)</p></blockquote>\n<p><strong>Scaling:</strong></p>\n<p>Stretching (steepening):  <em>a</em> ∙ ƒ(<em>x</em>) when <em>a</em> &gt; 1</p>\n<p>Compressing (flattening): <em>a</em> ∙ ƒ(<em>x</em>) when 0 &lt; <em>a</em> &lt; 1</p>\n"},{"title":"Sum of series","thumb":null,"image":null,"content":"<p>A series is the sum of terms in a sequence. The sum can be some of the terms or all of them. Use the formulas rather than listing the terms and adding them together.</p>\n<p>Arithmetic:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295178\" src=\"https://www.dummies.com/wp-content/uploads/sum-of-series-arithmetic.png\" alt=\"Sn =n/2 [2a1 +(n-1)d] = n/2(a1 + an)\" width=\"269\" height=\"47\" /></p>\n<p>Geometric:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295177\" src=\"https://www.dummies.com/wp-content/uploads/sum-of-series-geometric.png\" alt=\"Sn = g1(1-rn)/1-r\" width=\"117\" height=\"52\" /></p>\n<p>First <em>n</em> squares of the positive integers:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295176\" src=\"https://www.dummies.com/wp-content/uploads/sum-of-series-first-n-squares-of-positive-integers.png\" alt=\"1² + 2² +3² + L + n² = n(n+1)(2n+1)/6\" width=\"291\" height=\"50\" /></p>\n<p>First <em>n</em> cubes of the positive integers:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295179\" src=\"https://www.dummies.com/wp-content/uploads/sum-of-series-first-n-cubes-of-positive-integers.png\" alt=\"1³ + 2³ + 3³ + L + n³ = n²(n + 1)² / 4\" width=\"245\" height=\"47\" /></p>\n<p>First <em>n</em> odd positive integers:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-295180\" src=\"https://www.dummies.com/wp-content/uploads/sum-of-series-first-n-odd-positive-integers.png\" alt=\"1 + 3 + 5 + 7 + ... + (2n - 1) = n²\" width=\"235\" height=\"26\" /></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":"2022-10-04T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":294686},{"headers":{"creationTime":"2016-03-26T12:43:11+00:00","modifiedTime":"2022-09-22T12:27:57+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II: Making Matrices Work for You","strippedTitle":"algebra ii: making matrices work for you","slug":"algebra-ii-making-matrices-work-for-you","canonicalUrl":"","seo":{"metaDescription":"A matrix is a rectangular array of numbers. Each row has the same number of elements, and each column has the same number of elements. Matrices can be classifie","noIndex":0,"noFollow":0},"content":"<p>A <i>matrix</i> is a rectangular array of numbers. Each row has the same number of elements, and each column has the same number of elements. Matrices can be classified as: square, identity, zero, column, and so on.</p>\r\n<p>Where did matrices come from? For most of their history, they were called <i>arrays</i><i>.</i> There are references to <i>arrays</i> in Chinese, French, Italian, and many other mathematical works going back many hundreds of years. American mathematician George Dantzig's work with matrices during World War II allowed for the coordination of shipments of supplies and troops to various locations.</p>\r\n<p>Matrices are here to stay. You may be familiar with a method used to solve systems of linear equations using matrices, but this application just scratches the surface of what matrices can do.</p>\r\n<p>First, just in case you're <i>not</i> familiar with solving equations using matrices, let me give just a quick description. If you want to solve the following system of equations:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/425957.image0.png\" width=\"120\" height=\"70\" alt=\"image0.png\"/>\r\n<p>You write the matrix:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/425958.image1.png\" width=\"139\" height=\"70\" alt=\"image1.png\"/>\r\n<p>And then you perform row operations until you get the matrix:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/425959.image2.png\" width=\"111\" height=\"70\" alt=\"image2.png\"/>\r\n<p>From that matrix, you know that the solution of the system of equations is <i>x</i> = 1, <i>y</i> = -3, and <i>z</i> = -5. Pretty slick, don't you think?</p>\r\n<p>But uses for matrices don't stop there. You can solve traffic control problems, transportation logistics problems (how much of each item to send to various distribution centers), dietary problems (how much of each food product is needed to meet several different dietary requirements), and so on. Matrices work well in graphing calculators and computer spreadsheets — just set up the problem and let the technology do all the work.</p>","description":"<p>A <i>matrix</i> is a rectangular array of numbers. Each row has the same number of elements, and each column has the same number of elements. Matrices can be classified as: square, identity, zero, column, and so on.</p>\r\n<p>Where did matrices come from? For most of their history, they were called <i>arrays</i><i>.</i> There are references to <i>arrays</i> in Chinese, French, Italian, and many other mathematical works going back many hundreds of years. American mathematician George Dantzig's work with matrices during World War II allowed for the coordination of shipments of supplies and troops to various locations.</p>\r\n<p>Matrices are here to stay. You may be familiar with a method used to solve systems of linear equations using matrices, but this application just scratches the surface of what matrices can do.</p>\r\n<p>First, just in case you're <i>not</i> familiar with solving equations using matrices, let me give just a quick description. If you want to solve the following system of equations:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/425957.image0.png\" width=\"120\" height=\"70\" alt=\"image0.png\"/>\r\n<p>You write the matrix:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/425958.image1.png\" width=\"139\" height=\"70\" alt=\"image1.png\"/>\r\n<p>And then you perform row operations until you get the matrix:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/425959.image2.png\" width=\"111\" height=\"70\" alt=\"image2.png\"/>\r\n<p>From that matrix, you know that the solution of the system of equations is <i>x</i> = 1, <i>y</i> = -3, and <i>z</i> = -5. Pretty slick, don't you think?</p>\r\n<p>But uses for matrices don't stop there. You can solve traffic control problems, transportation logistics problems (how much of each item to send to various distribution centers), dietary problems (how much of each food product is needed to meet several different dietary requirements), and so on. Matrices work well in graphing calculators and computer spreadsheets — just set up the problem and let the technology do all the work.</p>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> taught mathematics for more than 45 years. She was a professor of mathematics at Bradley University for 35 of those years and continues to teach occasional classes either in person or via distance learning. Sterling is the author of several Dummies algebra and higher-level math titles. She is a graduate of the University of New Hampshire with a master's degree in math education.</p>","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"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":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-632ca2de9378c\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-632ca2de9486c\"></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":153053},{"headers":{"creationTime":"2016-03-26T21:48:01+00:00","modifiedTime":"2022-08-17T16:01:16+00:00","timestamp":"2022-09-14T18:19:56+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"How to Write Numbers in Scientific Notation","strippedTitle":"how to write numbers in scientific notation","slug":"how-to-write-numbers-in-scientific-notation","canonicalUrl":"","seo":{"metaDescription":"Learn how to write very large and very small numbers in scientific notation with these step-by-step instructions.","noIndex":0,"noFollow":0},"content":"<a href=\"https://www.dummies.com/education/math/pre-algebra/why-scientific-notation-works/\" target=\"_blank\" rel=\"noopener\">Scientific notation</a> is a standard way of writing very large and very small numbers so that they're easier to both compare and use in computations. To write in scientific notation, follow the form\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165063.image0.png\" alt=\"image0.png\" width=\"104\" height=\"36\" />\r\n\r\nwhere <i>N </i>is a number between 1 and 10, but not 10 itself, and <i>a </i>is an integer (positive or negative number).\r\n\r\nYou move the decimal point of a number until the new form is a number from 1 <i>up to</i> 10 (<i>N</i>), and then record the <a href=\"https://www.dummies.com/education/math/algebra/using-exponents-to-simplify-equations/\" target=\"_blank\" rel=\"noopener\">exponent</a> (<i>a</i>) as the number of places the decimal point was moved. Whether the power of 10 is positive or negative depends on whether you move the decimal to the right or to the left. Moving the decimal to the right makes the exponent negative; moving it to the left gives you a positive exponent.\r\nTo see an exponent that's positive, write <b>312,000,000,000</b> in scientific notation:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Move the decimal place to the <i>left</i> to create a new number from 1 up to 10.</p>\r\n<p class=\"child-para\">Where's the decimal point in 312,000,000,000? Because it's a whole number, the decimal point is <i>understood</i> to be at the end of the number: 312,000,000,000<b>.</b></p>\r\n<p class=\"child-para\">So, <i>N</i> = 3.12.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Determine the exponent, which is the number of times you moved the decimal.</p>\r\n<p class=\"child-para\">In this example, you moved the decimal 11 times; also, because you moved the decimal to the left, the exponent is positive. Therefore, <i>a</i> = 11, and so you get</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165064.image1.png\" alt=\"image1.png\" width=\"42\" height=\"32\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Put the number in the correct form for scientific notation</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165065.image2.png\" alt=\"image2.png\" width=\"310\" height=\"76\" /></li>\r\n</ol>\r\nTo see an exponent that's negative, write <b>.00000031</b> in scientific notation.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Move the decimal place to the <i>right</i> to create a new number from 1 up to 10.</p>\r\n<p class=\"child-para\">So, <i>N</i> = 3.1.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Determine the exponent, which is the number of times you moved the decimal.</p>\r\n<p class=\"child-para\">In this example, you moved the decimal 7 times; also, because you moved the decimal to the right, the exponent is negative. Therefore, <i>a</i> = –7, and so you get</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165066.image3.png\" alt=\"image3.png\" width=\"46\" height=\"32\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Put the number in the correct form for scientific notation</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165067.image4.png\" alt=\"image4.png\" width=\"248\" height=\"72\" /></li>\r\n</ol>\r\nWhen you get used to writing numbers in scientific notation, you can do it all in one step. Here are a few examples:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/437154.image0.jpg\" alt=\"image0.jpg\" width=\"265\" height=\"120\" />\r\n<h2 id=\"tab1\" >Order of magnitude</h2>\r\n<p class=\"TechnicalStuff\">Why does scientific notation always use a decimal between 1 and 10? The answer has to do with <i>order of magnitude</i>, which is a simple way to keep track of roughly how large a number is so you can compare numbers more easily. The order of magnitude of a number is its exponent in scientific notation. For example,</p>\r\n<p class=\"article-list\">703 = 7.03 x 10<sup>2</sup> — order of magnitude is 2\r\n600,000 = 6 x 10<sup>5</sup> — order of magnitude is 5\r\n0.00095 = 9.5 x 10<sup>–4</sup> — order of magnitude is –4</p>\r\nEvery number between 10 and 100 has an order of magnitude of 1. Every number between 100 and 1,000 has an order of magnitude of 2.","description":"<a href=\"https://www.dummies.com/education/math/pre-algebra/why-scientific-notation-works/\" target=\"_blank\" rel=\"noopener\">Scientific notation</a> is a standard way of writing very large and very small numbers so that they're easier to both compare and use in computations. To write in scientific notation, follow the form\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165063.image0.png\" alt=\"image0.png\" width=\"104\" height=\"36\" />\r\n\r\nwhere <i>N </i>is a number between 1 and 10, but not 10 itself, and <i>a </i>is an integer (positive or negative number).\r\n\r\nYou move the decimal point of a number until the new form is a number from 1 <i>up to</i> 10 (<i>N</i>), and then record the <a href=\"https://www.dummies.com/education/math/algebra/using-exponents-to-simplify-equations/\" target=\"_blank\" rel=\"noopener\">exponent</a> (<i>a</i>) as the number of places the decimal point was moved. Whether the power of 10 is positive or negative depends on whether you move the decimal to the right or to the left. Moving the decimal to the right makes the exponent negative; moving it to the left gives you a positive exponent.\r\nTo see an exponent that's positive, write <b>312,000,000,000</b> in scientific notation:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Move the decimal place to the <i>left</i> to create a new number from 1 up to 10.</p>\r\n<p class=\"child-para\">Where's the decimal point in 312,000,000,000? Because it's a whole number, the decimal point is <i>understood</i> to be at the end of the number: 312,000,000,000<b>.</b></p>\r\n<p class=\"child-para\">So, <i>N</i> = 3.12.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Determine the exponent, which is the number of times you moved the decimal.</p>\r\n<p class=\"child-para\">In this example, you moved the decimal 11 times; also, because you moved the decimal to the left, the exponent is positive. Therefore, <i>a</i> = 11, and so you get</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165064.image1.png\" alt=\"image1.png\" width=\"42\" height=\"32\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Put the number in the correct form for scientific notation</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165065.image2.png\" alt=\"image2.png\" width=\"310\" height=\"76\" /></li>\r\n</ol>\r\nTo see an exponent that's negative, write <b>.00000031</b> in scientific notation.\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Move the decimal place to the <i>right</i> to create a new number from 1 up to 10.</p>\r\n<p class=\"child-para\">So, <i>N</i> = 3.1.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Determine the exponent, which is the number of times you moved the decimal.</p>\r\n<p class=\"child-para\">In this example, you moved the decimal 7 times; also, because you moved the decimal to the right, the exponent is negative. Therefore, <i>a</i> = –7, and so you get</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165066.image3.png\" alt=\"image3.png\" width=\"46\" height=\"32\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Put the number in the correct form for scientific notation</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165067.image4.png\" alt=\"image4.png\" width=\"248\" height=\"72\" /></li>\r\n</ol>\r\nWhen you get used to writing numbers in scientific notation, you can do it all in one step. Here are a few examples:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/437154.image0.jpg\" alt=\"image0.jpg\" width=\"265\" height=\"120\" />\r\n<h2 id=\"tab1\" >Order of magnitude</h2>\r\n<p class=\"TechnicalStuff\">Why does scientific notation always use a decimal between 1 and 10? The answer has to do with <i>order of magnitude</i>, which is a simple way to keep track of roughly how large a number is so you can compare numbers more easily. The order of magnitude of a number is its exponent in scientific notation. For example,</p>\r\n<p class=\"article-list\">703 = 7.03 x 10<sup>2</sup> — order of magnitude is 2\r\n600,000 = 6 x 10<sup>5</sup> — order of magnitude is 5\r\n0.00095 = 9.5 x 10<sup>–4</sup> — order of magnitude is –4</p>\r\nEvery number between 10 and 100 has an order of magnitude of 1. Every number between 100 and 1,000 has an order of magnitude of 2.","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <b>Mark Zegarelli</b> is a professional writer with degrees in both English and Math from Rutgers University. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. Along the way, he&#8217;s also paid a few bills doing housecleaning, decorative painting, and (for ten hours) retail sales. He likes writing best, though.","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Order of magnitude","target":"#tab1"}],"relatedArticles":{"fromBook":[{"articleId":158569,"title":"Inverse Operations and Commutative, Associative, and Distributive Properties","slug":"inverse-operations-and-commutative-associative-and-distributive-properties","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/158569"}},{"articleId":158567,"title":"Converting Metric Units to English Units","slug":"converting-metric-units-to-english-units","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/158567"}},{"articleId":158560,"title":"A Guide to Working with Exponents, Radicals, and Absolute Value","slug":"a-guide-to-working-with-exponents-radicals-and-absolute-value","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/158560"}},{"articleId":158557,"title":"A Quick Conversion Guide for Fractions, Decimals, and Percents","slug":"a-quick-conversion-guide-for-fractions-decimals-and-percents","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/158557"}},{"articleId":158554,"title":"Following the Order of Operations","slug":"following-the-order-of-operations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/158554"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281978,"slug":"basic-math-pre-algebra-for-dummies-2nd-edition","isbn":"9781119293637","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"https://www.amazon.com/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119293634/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/1119293634-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/basic-math-and-pre-algebra-for-dummies-2nd-edition-cover-9781119293637-203x255.jpg","width":203,"height":255},"title":"Basic Math & Pre-Algebra For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><b><b data-author-id=\"9399\">Mark Zegarelli</b></b> is a professional writer with degrees in both English and Math from Rutgers University. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. Along the way, he&#8217;s also paid a few bills doing housecleaning, decorative painting, and (for ten hours) retail sales. He likes writing best, though.</p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <b>Mark Zegarelli</b> is a professional writer with degrees in both English and Math from Rutgers University. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. Along the way, he&#8217;s also paid a few bills doing housecleaning, decorative painting, and (for ten hours) retail sales. He likes writing best, though.","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9399"}}],"_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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-63221b4c813e6\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-63221b4c81e3a\"></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":"2022-08-17T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":194384},{"headers":{"creationTime":"2016-03-27T16:56:11+00:00","modifiedTime":"2022-04-28T15:29:39+00:00","timestamp":"2022-09-14T18:19:41+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra Workbook For Dummies Cheat Sheet","strippedTitle":"algebra workbook for dummies cheat sheet","slug":"algebra-workbook-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Use these formulas, patterns, and procedures to simplify expressions and solve equations for all your algebra needs.","noIndex":0,"noFollow":0},"content":"Formulas, patterns, and procedures used for simplifying expressions and solving equations are basic to algebra. Use the equations, shortcuts, and formulas you find for quick reference. This Cheat Sheet offers basic information and short explanations (and some words of advice on traps to avoid).","description":"Formulas, patterns, and procedures used for simplifying expressions and solving equations are basic to algebra. Use the equations, shortcuts, and formulas you find for quick reference. This Cheat Sheet offers basic information and short explanations (and some words of advice on traps to avoid).","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":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281939,"slug":"algebra-i-workbook-for-dummies-3rd-edition","isbn":"9781119348955","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119348951/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119348951/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/1119348951-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119348951/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119348951/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/algebra-i-workbook-for-dummies-3rd-edition-cover-9781119348955-204x255.jpg","width":204,"height":255},"title":"Algebra I 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 math-ematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of <i>Algebra I For Dummies</i> and <i>Algebra II 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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119348955&quot;]}]\" id=\"du-slot-63221b3e000fd\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119348955&quot;]}]\" id=\"du-slot-63221b3e00b74\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":192380,"title":"Order of Operations in Algebra","slug":"order-of-operations-in-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/192380"}},{"articleId":192379,"title":"Rules of Exponents","slug":"rules-of-exponents","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/192379"}},{"articleId":192386,"title":"Selected Math Formulas Step by Step","slug":"selected-math-formulas-step-by-step","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/192386"}},{"articleId":192387,"title":"Factoring Special Problems","slug":"factoring-special-problems","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/192387"}},{"articleId":192383,"title":"Formulas for Common Geometric Shapes","slug":"formulas-for-common-geometric-shapes","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/192383"}}],"content":[{"title":"Order of operations in algebra","thumb":null,"image":null,"content":"<p>When creating simpler and more useful expressions, you want to be careful not to change the original value. By applying the order of operations, you maintain that value.</p>\n<p>Apply the order of operations when no grouping symbols, such as parentheses, interrupt. When more of one level occurs in a problem, do them in order from left to right. When you perform operations on algebraic expressions and you have a choice between one or more operations to perform, use the following order:</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">Powers and roots</p>\n</li>\n<li>\n<p class=\"first-para\">Multiplication and division</p>\n</li>\n<li>\n<p class=\"first-para\">Addition and subtraction</p>\n</li>\n</ol>\n<p>These rules are interrupted if the problem has grouping symbols. You first need to perform operations in grouping symbols, such as ( ), { }, [ ] , above and below fraction lines, and inside radicals.</p>\n"},{"title":"Rules of exponents","thumb":null,"image":null,"content":"<p>Exponents are shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base to something simpler. Remember that in xa the x is the base and the “a�? is the exponent.</p>\n<p>Assume that neither x nor y are equal to zero:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200147.image0.jpg\" alt=\"image0.jpg\" width=\"171\" height=\"400\" /></p>\n"},{"title":"Selected math formulas step by step","thumb":null,"image":null,"content":"<p>Algebraic formulas make life (and algebra) simpler. You save time by not having to perform more complicated tasks. When using the formulas, use the appropriate rules for simplifying algebraic expressions. Also watch out for pitfalls; to help you, an asterisk (*) appears beside steps where errors are easy to make.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200149.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"255\" /><br />\n<img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200150.image1.jpg\" alt=\"image1.jpg\" width=\"535\" height=\"655\" /></p>\n"},{"title":"Factoring special problems","thumb":null,"image":null,"content":"<p>Binomials, their powers, and their products with selected trinomials occur frequently in algebraic processes. By using the patterns shown here, you save time and reduce the opportunity for errors.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/200143.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"274\" /></p>\n"},{"title":"Formulas for common geometric shapes","thumb":null,"image":null,"content":"<p>Depending on the algebra problem, you&#8217;ll need to know some geometry. The following represents some of the most common shapes in geometry and their formulas for perimeter, area, volume, surface areas, and circumference:</p>\n<table border=\"0\">\n<caption>Two-Dimensional Shapes</caption>\n<tbody>\n<tr>\n<th>Shape</th>\n<th>Perimeter/Circumference</th>\n<th>Area</th>\n</tr>\n<tr>\n<td>Rectangle</td>\n<td>P = 2(l + w)</td>\n<td>A = lw</td>\n</tr>\n<tr>\n<td>Square</td>\n<td>P = 4s</td>\n<td>A = s<sup>2</sup></td>\n</tr>\n<tr>\n<td>Triangle</td>\n<td>P = a + b + c</td>\n<td>A = 1/2bh</td>\n</tr>\n<tr>\n<td>Trapezoid</td>\n<td>P = a + b<sub>1</sub> + c + b<sub>2</sub></td>\n<td>A = 1/2h(b<sub>1</sub> + b<sub>2</sub>)</td>\n</tr>\n<tr>\n<td>Isosceles Trapezoid</td>\n<td>P = 2w + b<sub>1</sub> + b<sub>2</sub></td>\n<td>A = 1/2h(b<sub>1</sub> + b<sub>2</sub>)</td>\n</tr>\n<tr>\n<td>Circle</td>\n<td>C = πd = 2 π r</td>\n<td>A = π r<sup>2</sup></td>\n</tr>\n</tbody>\n</table>\n<table border=\"0\">\n<caption>Three-Dimensional Shapes</caption>\n<tbody>\n<tr>\n<th>Shape</th>\n<th>Surface Area</th>\n<th>Volume</th>\n</tr>\n<tr>\n<td>Box</td>\n<td>SA = 2lw + 2lh + 2wh</td>\n<td>V = lwh</td>\n</tr>\n<tr>\n<td>Sphere</td>\n<td>SA = 4 π r<sup>2</sup></td>\n<td>V = 4/3 π r<sup>3</sup></td>\n</tr>\n<tr>\n<td>Cylinder</td>\n<td>SA = 2 π r(r + h)</td>\n<td>V = π r<sup>2</sup>h</td>\n</tr>\n</tbody>\n</table>\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":"2022-04-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":209000},{"headers":{"creationTime":"2016-03-27T16:49:18+00:00","modifiedTime":"2022-04-19T20:51:04+00:00","timestamp":"2022-09-14T18:19:38+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II: 1001 Practice Problems For Dummies Cheat Sheet","strippedTitle":"algebra ii: 1001 practice problems for dummies cheat sheet","slug":"1001-algebra-ii-practice-problems-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This handy Cheat Sheet is a great reference for algebra problems, including equations, systems, graph lines, functions, and more.","noIndex":0,"noFollow":0},"content":"The best way to figure out how the different algebraic rules work and interact with one another is to practice with lots of problems. And Algebra II requires lots of practice. So be prepared to solve equations and systems, graph lines, tackle functions, and so much more.","description":"The best way to figure out how the different algebraic rules work and interact with one another is to practice with lots of problems. And Algebra II requires lots of practice. So be prepared to solve equations and systems, graph lines, tackle functions, and so much more.","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":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":165288,"title":"Working with Radical and Rational Equations in Algebra II","slug":"working-with-radical-and-rational-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165288"}},{"articleId":165267,"title":"Polynomial Functions and Equations in Algebra II","slug":"polynomial-functions-and-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165267"}},{"articleId":165264,"title":"Systems of Linear Equations in Algebra II","slug":"systems-of-linear-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165264"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281912,"slug":"algebra-ii-1001-practice-problems-for-dummies-free-online-practice","isbn":"9781119883562","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119883563/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119883563/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/1119883563-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119883563/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119883563/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/9781119883562-204x255.jpg","width":204,"height":255},"title":"Algebra II: 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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883562&quot;]}]\" id=\"du-slot-63221b3aa794b\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883562&quot;]}]\" id=\"du-slot-63221b3aa8219\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":165288,"title":"Working with Radical and Rational Equations in Algebra II","slug":"working-with-radical-and-rational-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165288"}},{"articleId":165267,"title":"Polynomial Functions and Equations in Algebra II","slug":"polynomial-functions-and-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165267"}},{"articleId":165264,"title":"Systems of Linear Equations in Algebra II","slug":"systems-of-linear-equations-in-algebra-ii","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/165264"}}],"content":[{"title":"Working with radical and rational equations","thumb":null,"image":null,"content":"<p>A <i>radical</i> equation is one that starts out with a square root, cube root, or some other root and gets changed into another form to make the solving process easier. A <i>rational </i>equation is one that involves a fractional expression — usually with a polynomial in the numerator and denominator. Avoid these mistakes when working with radical or rational equations:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Forgetting to check for extraneous solutions</p>\n</li>\n<li>\n<p class=\"first-para\">Squaring a binomial incorrectly when squaring both sides to get rid of the radical</p>\n</li>\n<li>\n<p class=\"first-para\">Distributing correctly when writing equivalent fractions using a common denominator</p>\n</li>\n<li>\n<p class=\"first-para\">Eliminating solutions that create a 0 in the denominator</p>\n</li>\n</ul>\n"},{"title":"Polynomial functions and equations","thumb":null,"image":null,"content":"<p>In Algebra II, a polynomial function is one in which the coefficients are all real numbers, and the exponents on the variables are all whole numbers. A polynomial whose greatest power is 2 is called a <i>quadratic polynomial</i><i>;</i> if the highest power is 3, then it’s called a <i>cubic polynomial</i><i>.</i> A highest power of 4 earns the name <i>quartic</i> (not to be confused with quadratic), and a highest power of 5 is called <i>quintic</i><i>.</i></p>\n<p>When solving polynomial functions and equations, don’t let these common mistakes trip you up:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Forgetting to change the signs in the factored form when identifying x-intercepts</p>\n</li>\n<li>\n<p class=\"first-para\">Making errors when simplifying the terms in f(–x) applying Descartes’ rule of sign</p>\n</li>\n<li>\n<p class=\"first-para\">Not changing the sign of the divisor when using synthetic division</p>\n</li>\n<li>\n<p class=\"first-para\">Not distinguishing between curves that cross from those that just touch the x-axis at an intercept</p>\n</li>\n<li>\n<p class=\"first-para\">Graphing the incorrect end-behavior on the right and left of the graphs</p>\n</li>\n</ul>\n"},{"title":"Systems of linear equations","thumb":null,"image":null,"content":"<p>In Algebra II, a <i>linear equation</i> consists of variable terms whose exponents are always the number 1. When you have two variables, the equation can be represented by a line. With three terms, you can draw a plane to describe the equation. More than three variables is indescribable, because there are only three dimensions. When you have a system of linear equations, you can find the values of the variables that work for all the equations in the system — the common solutions. Sometimes there’s just one solution, sometimes many, and sometimes there’s no solution at all.</p>\n<p>When solving systems of linear equations, watch out for these mistakes:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Forgetting to change the signs in the factored form when identifying x-intercepts</p>\n</li>\n<li>\n<p class=\"first-para\">Making errors when simplifying the terms in f(–x) applying Descartes’ rule of sign</p>\n</li>\n<li>\n<p class=\"first-para\">Not changing the sign of the divisor when using synthetic division</p>\n</li>\n<li>\n<p class=\"first-para\">Not distinguishing between curves that cross from those that just touch the x-axis at an intercept</p>\n</li>\n<li>\n<p class=\"first-para\">Graphing the incorrect end-behavior on the right and left of the graphs</p>\n</li>\n</ul>\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":207944},{"headers":{"creationTime":"2016-03-27T16:48:03+00:00","modifiedTime":"2022-04-12T18:48:58+00:00","timestamp":"2022-09-14T18:19:36+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II Workbook For Dummies Cheat Sheet","strippedTitle":"algebra ii workbook for dummies cheat sheet","slug":"algebra-ii-workbook-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Learning some algebraic rules for various exponents, formulas, and equations will help you successfully solve problems in Algebra II.","noIndex":0,"noFollow":0},"content":"Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.","description":"Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.","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":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":149411,"title":"Solving Equations with Complex Solutions","slug":"solving-equations-with-complex-solutions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149411"}},{"articleId":149344,"title":"How to Put Equations of Parabolas in Standard Form","slug":"how-to-put-equations-of-parabolas-in-standard-form","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/149344"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281941,"slug":"algebra-ii-workbook-for-dummies-3rd-edition","isbn":"9781119543114","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119543118/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119543118/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/1119543118-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119543118/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119543118/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/algebra-ii-workbook-for-dummies-3rd-edition-cover-9781119543114-204x255.jpg","width":204,"height":255},"title":"Algebra II Workbook For Dummies","testBankPinActivationLink":"https://testbanks.wiley.com","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> taught junior high and high school math before embarking on her career as an instructor at Bradley University, where she taught for more than 35 years. </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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543114&quot;]}]\" id=\"du-slot-63221b3868c33\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119543114&quot;]}]\" id=\"du-slot-63221b38695be\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":153121,"title":"Algebra: Rules of Exponents","slug":"algebra-rules-of-exponents","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153121"}},{"articleId":153120,"title":"Linear Equations: How to Find Slope, <i>y</i>-Intercept, Distance, Midpoint","slug":"linear-equations-how-to-find-slope-y-intercept-distance-midpoint","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153120"}},{"articleId":153075,"title":"Rewrite Absolute Value Equations as Linear Equations","slug":"rewrite-absolute-value-equations-as-linear-equations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153075"}},{"articleId":153020,"title":"9 Number Systems in Algebra to Know","slug":"9-number-systems-in-algebra-to-know","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153020"}},{"articleId":153123,"title":"Algebra II: What Is the Binomial Theorem?","slug":"algebra-ii-what-is-the-binomial-theorem","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153123"}},{"articleId":153137,"title":"Use the Properties of Proportions to Simplify Fractions","slug":"use-the-properties-of-proportions-to-simplify-fractions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153137"}},{"articleId":153138,"title":"Algebra II: Raise Binomials to a Power","slug":"algebra-ii-raise-binomials-to-a-power","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/153138"}}],"content":[{"title":"Algebra: Rules of exponents","thumb":null,"image":null,"content":"<p>Exponents are shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base into something simpler to work with. Remember that in <i>x</i><i><sup>a</sup></i>, the <i>x</i> is the <i>base</i> and the <i>a</i> is the <i>exponent</i>.</p>\n<p>Assume <i>x</i> ≠ 0:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425924.image0.png\" alt=\"image0.png\" width=\"73\" height=\"202\" /></p>\n"},{"title":"Linear equations: how to find slope, y-intercept, distance, midpoint","thumb":null,"image":null,"content":"<p>In algebra, linear equations means you&#8217;re dealing with straight lines. When you&#8217;re working with the <i>xy</i>-coordinate system, you can use the following formulas to find the slope, <i>y</i>-intercept, distance, and midpoint between two points.</p>\n<p>Consider the two points (<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>) and (<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>):</p>\n<blockquote><p><b>Slope of the line through the points:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425926.image0.png\" alt=\"image0.png\" width=\"72\" height=\"30\" /></p>\n<blockquote><p><b>Slope-intercept form of the line with <i>y</i>-intercept <i>b</i>:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425927.image1.png\" alt=\"image1.png\" width=\"68\" height=\"15\" /></p>\n<blockquote><p><b>Point-slope form of the line with slope <i>m</i>:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425928.image2.png\" alt=\"image2.png\" width=\"110\" height=\"15\" /></p>\n<blockquote><p><b>Distance formula:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425929.image3.png\" alt=\"image3.png\" width=\"168\" height=\"28\" /></p>\n<blockquote><p><b>Midpoint formula:</b></p></blockquote>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425930.image4.png\" alt=\"image4.png\" width=\"140\" height=\"27\" /></p>\n"},{"title":"Rewrite absolute value equations as linear equations","thumb":null,"image":null,"content":"<p>To work with an absolute value equation in algebra, you first need to rewrite it as a linear equation. The same goes for an absolute value inequality, which you rewrite as a linear inequality.</p>\n<p>When rewriting absolute value equations or inequalities, you drop the absolute value bars.</p>\n<blockquote><p>|<i>ax</i> + <i>b</i>| = <i>c</i><i> </i><i>→</i> <i>ax</i> + <i>b</i> = <i>c</i> or <i>ax</i> + <i>b</i> = –<i>c</i></p></blockquote>\n<blockquote><p>|<i>ax</i> + <i>b</i>| &gt; <i>c →</i> <i>ax</i> + <i>b</i> &gt; <i>c</i> or <i>ax</i> + <i>b</i> &gt; –<i>c</i></p></blockquote>\n<blockquote><p>|<i>ax</i> + <i>b</i>| &lt; <i>c →</i> –<i>c</i> &lt; <i>ax</i> + <i>b</i> &lt; <i>c</i></p></blockquote>\n"},{"title":"9 number systems in algebra to know","thumb":null,"image":null,"content":"<p>A <i>number system</i> in algebra is a set of numbers — and different number systems are used to solve different types of algebra problems. Number systems include real numbers, natural numbers, whole numbers, integers, rational numbers, irrational numbers, even numbers, and odd numbers.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Real numbers: </b>Real numbers comprise the full spectrum of numbers. They cover the gamut and can take on any form — fractions or whole numbers, decimal points or no decimal points. The full range of real numbers includes decimals that can go on forever. Real numbers are different from imaginary or complex numbers.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Natural numbers:</b> A natural number is a number that comes naturally. What numbers did you first use? Remember someone asking, &#8220;How old are you?&#8221; You proudly held up four fingers and said, &#8220;Four!&#8221; Natural numbers are greater than zero but don&#8217;t include fractions: 1, 2, 3, 4, 5, 6, 7, and so on, into infinity. You use natural numbers to count items and to make lists.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Whole numbers: </b>Whole numbers are just all the natural numbers plus a zero: 0, 1, 2, 3, 4, 5, and so on, into infinity. They act like natural numbers and are used when whole amounts (no fractions) are required. Zero can also indicate none.</p>\n<p class=\"child-para Remember\">Algebraic problems often require you to round the answer to the nearest whole number. This makes perfect sense when the problem involves people, cars, animals, houses, or anything that shouldn&#8217;t be cut into pieces.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Integers:</b> Integers incorporate all the whole numbers and their opposites (or additive inverses of the whole numbers). Integers can be described as being positive and negative whole numbers and 0: . . . –3, –2, –1, 0, 1, 2, 3, . . . .</p>\n<p class=\"child-para\">Integers are popular in algebra. When you solve a long, complicated problem and come up with an integer, you can be joyous because your answer is probably right. After all, it&#8217;s not a fraction! This doesn&#8217;t mean that answers in algebra can&#8217;t be fractions or decimals. It&#8217;s just that most textbooks and reference books try to stick with nice answers to increase the comfort level and avoid confusion.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Rational numbers: </b>Rational numbers are numbers that act rationally! In this case, acting rationally means that the decimal equivalent of the rational number behaves. The decimal ends somewhere, or it has a repeating pattern to it. That&#8217;s what constitutes &#8220;behaving.&#8221; Some examples of rational numbers with decimals that terminate include 2, 3.4, 5.77623, and –4.5.</p>\n<p class=\"child-para\">Some examples of rational numbers with decimals that repeat the same pattern include the following:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425933.image0.png\" alt=\"image0.png\" width=\"88\" height=\"15\" /><br />\n<img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425934.image1.png\" alt=\"image1.png\" width=\"88\" height=\"15\" /></p>\n<p class=\"child-para\">(The horizontal bar over the 164 and the 6 lets you know that these numbers repeat forever.) In all cases, rational numbers can be written as a fraction. They all have a fraction that they are equal to.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Irrational numbers:</b> Irrational numbers are real numbers that are not rational numbers. An irrational number cannot be written as a fraction, and decimal values for irrationals never end and never have a nice pattern to them. For example, pi, with its never-ending decimal places, is irrational.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Imaginary/complex numbers: </b>A number that isn&#8217;t real can be imaginary or complex. An imaginary number contains some multiple of <i>i</i>, which is the following:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425935.image2.png\" alt=\"image2.png\" width=\"34\" height=\"18\" /></p>\n<p class=\"child-para\">For example, 2 + 3<i>i</i> is a complex number.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Even numbers:</b> An even number is one that divides evenly by 2, such as 2, 4, 18, and 352.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Odd numbers:</b> An odd number is one that does not divide evenly by 2, such as 1, 3, 27, and 485.</p>\n</li>\n</ul>\n"},{"title":"Algebra II: What is the binomial theorem?","thumb":null,"image":null,"content":"<p>A <i>binomial</i> is a mathematical expression that has two terms. In algebra, people frequently raise binomials to powers to complete computations. The binomial theorem says that if <i>a</i> and <i>b</i> are real numbers and <i>n</i> is a positive integer, then</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425937.image0.png\" alt=\"image0.png\" width=\"507\" height=\"83\" /></p>\n<p>You can see the rule here, in the second line, in terms of the coefficients that are created using combinations. The powers on <i>a</i> start with <i>n</i> and decrease until the power is zero in the last term. That&#8217;s why you don&#8217;t see an <i>a</i> in the last term — it&#8217;s <i>a</i><sup>0</sup>, which is really a 1. The powers on <i>b</i> increase from <i>b</i><sup>0 </sup>until the last term, where it&#8217;s <i>b</i><i><sup>n</sup></i>. Notice that the power of <i>b</i> matches <i>k</i> in the combination.</p>\n"},{"title":"Use the properties of proportions to simplify fractions","thumb":null,"image":null,"content":"<p>In algebra, the properties of proportions come in handy when solving equations involving fractions. When you can, change an algebraic equation with fractions in it to a proportion for easy solving.</p>\n<p>If</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425939.image0.png\" alt=\"image0.png\" width=\"42\" height=\"24\" /></p>\n<p>then the following are all true:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425940.image1.png\" alt=\"image1.png\" width=\"42\" height=\"28\" /></p>\n<p>&nbsp;</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425941.image2.png\" alt=\"image2.png\" width=\"85\" height=\"28\" /></p>\n<p>&nbsp;</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425942.image3.png\" alt=\"image3.png\" width=\"85\" height=\"28\" /></p>\n<p>&nbsp;</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425943.image4.png\" alt=\"image4.png\" width=\"85\" height=\"36\" /></p>\n<p>A proportion is an equation involving two ratios (fractions) set equal to each other. The following equation is a proportion:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425944.image5.png\" alt=\"image5.png\" width=\"57\" height=\"27\" /></p>\n<p>Both fractions in that proportion reduce to</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425945.image6.png\" alt=\"image6.png\" width=\"13\" height=\"27\" /></p>\n<p>so it&#8217;s fairly easy to see how this statement is true.</p>\n<p>Proportions have some interesting, helpful, and easy-to-use properties. For example, in the following proportion,</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425946.image7.png\" alt=\"image7.png\" width=\"42\" height=\"24\" /></p>\n<p>the cross-products are equal: <em>a</em> <em>∙</em> <em>d = b</em> <em>∙</em> <em>c</em>.</p>\n<p>The reciprocals are equal (you can flip the fractions):</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/425947.image8.png\" alt=\"image8.png\" width=\"42\" height=\"28\" /></p>\n<p>You can reduce the fractions vertically or horizontally: You can divide out factors that are common to both numerators or both denominators or the left fraction or the right fraction. (You can&#8217;t, however, divide out a factor from the numerator of one fraction and the denominator of the other.)</p>\n"},{"title":"Algebra II: Raise binomials to a power","thumb":null,"image":null,"content":"<p>A <i>binomial</i> is a mathematical expression that has two terms. In algebra, people frequently raise binomials to powers in order to solve equations. Here are some examples:</p>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>0</sup> = 1</p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>1</sup> = <i>a</i> + <i>b</i></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>2</sup> = <i>a</i><sup>2</sup> + 2<i>ab</i> + <i>b</i><sup>2</sup></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>3</sup> = <i>a</i><sup>3</sup> + 3<i>a</i><sup>2</sup><i>b</i> + 3<i>ab</i><sup>2</sup> + <i>b</i><sup>3</sup></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>4</sup> = <i>a</i><sup>4</sup> + 4<i>a</i><sup>3</sup><i>b</i> + 6<i>a</i><sup>2</sup><i>b</i><sup>2</sup> + 4<i>ab</i><sup>3</sup> + <i>b</i><sup>4</sup></p></blockquote>\n<blockquote><p>(<i>a</i> + <i>b</i>)<sup>5</sup> = <i>a</i><sup>5</sup> + 5<i>a</i><sup>4</sup><i>b</i> + 10<i>a</i><sup>3</sup><i>b</i><sup>2</sup> + 10<i>a</i><sup>2</sup><i>b</i><sup>3</sup> + 5<i>ab</i><sup>4</sup> + <i>b</i><sup>5</sup></p></blockquote>\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":"2022-04-12T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207689},{"headers":{"creationTime":"2016-03-27T16:49:35+00:00","modifiedTime":"2022-03-28T14:24:38+00:00","timestamp":"2022-09-14T18:19:31+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra I: 1001 Practice Problems For Dummies Cheat Sheet","strippedTitle":"algebra i: 1001 practice problems for dummies cheat sheet","slug":"1001-algebra-i-practice-problems-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"As you're learning algebra, keep this handy guide nearby as a reference for factoring and solving basic equations.","noIndex":0,"noFollow":0},"content":"Algebra problems are easier to solve when you know the rules and formulas. Unlike other subjects where you can just read or listen and absorb the information sufficiently, math takes practice.\r\n\r\nThe only way to figure out how the different algebraic rules work and interact with one another is to get into the problems and get your hands dirty, so to speak. Be prepared to practice on lots of different problems.","description":"Algebra problems are easier to solve when you know the rules and formulas. Unlike other subjects where you can just read or listen and absorb the information sufficiently, math takes practice.\r\n\r\nThe only way to figure out how the different algebraic rules work and interact with one another is to get into the problems and get your hands dirty, so to speak. Be prepared to practice on lots of different problems.","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":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":167671,"title":"Factoring in Algebra I","slug":"factoring-in-algebra-i","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167671"}},{"articleId":167670,"title":"The Basics of Solving Algebra I Equations","slug":"the-basics-of-solving-algebra-i-equations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167670"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281911,"slug":"algebra-i-1001-practice-problems-for-dummies-free-online-practice","isbn":"9781119883470","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119883474/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119883474/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/1119883474-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119883474/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119883474/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/9781119883470-204x255.jpg","width":204,"height":255},"title":"Algebra I: 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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883470&quot;]}]\" id=\"du-slot-63221b33c8016\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883470&quot;]}]\" id=\"du-slot-63221b33c8b26\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":167671,"title":"Factoring in Algebra I","slug":"factoring-in-algebra-i","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167671"}},{"articleId":167670,"title":"The Basics of Solving Algebra I Equations","slug":"the-basics-of-solving-algebra-i-equations","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/167670"}}],"content":[{"title":"Factoring in algebra","thumb":null,"image":null,"content":"<p>Factoring algebraic expressions is one of the most important techniques you need to practice. Not much else can be done in terms of solving equations, graphing functions and conics, and working on math applications if you can&#8217;t pull out a common factor and simplify an expression.</p>\n<p>Here are some tips for factoring in algebra:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Make sure you apply divisibility rules correctly.</p>\n</li>\n<li>\n<p class=\"first-para\">Write a prime factorization with the correct exponents on the prime factors.</p>\n</li>\n<li>\n<p class=\"first-para\">Check that the terms divided after dividing out a greatest common factor (GCF) don&#8217;t still have a common factor.</p>\n</li>\n<li>\n<p class=\"first-para\">Reduce only factors, not terms.</p>\n</li>\n<li>\n<p class=\"first-para\">Write fractional answers with correct grouping symbols to distinguish remaining factors.</p>\n</li>\n</ul>\n<h3>Factoring binomials</h3>\n<p>A binomial is an expression with two terms. The terms can be separated by addition or subtraction. You have four possibilities for factoring binomials:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Factor out a greatest common factor.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor as the difference of perfect squares.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor as the difference of perfect cubes.</p>\n</li>\n<li>\n<p class=\"first-para\">Factor as the sum of perfect cubes.</p>\n</li>\n</ul>\n<p>If one of these methods doesn&#8217;t work, then the binomial doesn&#8217;t factor by using real numbers.</p>\n<h3>Factoring quadratic trinomials</h3>\n<p>You can factor trinomials with the form <i>ax</i><sup>2</sup> + <i>bx</i> + <i>c</i> in one of two ways:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Factor out a greatest common factor.</p>\n</li>\n<li>\n<p class=\"first-para\">Find two binomials whose product is that trinomial.</p>\n</li>\n</ul>\n<p>When finding the two binomials whose product is a particular trinomial, you work from the factors of the constant term and the factors of the coefficient of the lead term to create a sum or difference that matches the coefficient of the middle term. This technique can be expanded to trinomials that have the same general format but with exponents that are multiples of the basic trinomial.</p>\n"},{"title":"The basics of solving algebra equations","thumb":null,"image":null,"content":"<p>One of the most common goals in algebra I is solve an equation. Solving an equation means to identify the number or numbers you can replace the variable with to make a true statement. You&#8217;ll find factoring and the multiplication property of zero to be your first approach, and then you&#8217;ll also have the quadratic formula to use on some of the more challenging second degree equations. Polynomials can be solved using synthetic division to help with the factoring.</p>\n<h3>Linear equations</h3>\n<p>Linear equations are of the form <i>ax</i> + <i>b</i> = <i>c</i><i>,</i> where <i>x</i> is some variable, and <i>a</i><i>,</i> <i>b</i><i>,</i> and <i>c</i> are real numbers. To solve a linear equation, you perform a series of <i>opposite</i><i>s</i><i>:</i></p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">If a number is added to the term containing <i>x</i><i>,</i> you subtract that number from both sides of the equation.</p>\n</li>\n<li>\n<p class=\"first-para\">If a number is subtracted from the term containing the variable, you add.</p>\n</li>\n<li>\n<p class=\"first-para\">If a number multiplies the variable, you divide.</p>\n</li>\n<li>\n<p class=\"first-para\">If a number divides the variable, you multiply.</p>\n</li>\n</ul>\n<p>Just be sure that whatever you do to one side of the equation, you also do to the other side. Think of the equation as two expressions pivoting on either side of a balance scale: You need to keep the sides at the same <i>weigh</i><i>t</i><i>.</i></p>\n<h3>Quadratic equations</h3>\n<p>A quadratic equation has the form <i>ax</i><sup>2</sup> + <i>bx</i> + <i>c</i> = 0. The equation can have two solutions, only one solution (a double root), or no solutions among the real numbers. Where no real solution occurs, imaginary numbers are brought into the picture. Quadratic equations are solved most easily when the trinomial factors, but the quadratic formula is also a nice means to finding solutions.</p>\n<h3>Polynomials</h3>\n<p>A polynomial is a smooth curve that goes on and on forever, from negative infinity to positive infinity. To solve a polynomial means to set the equation equal to 0 and determine which, if any, numbers create a true statement. Any numbers satisfying this equation give you important information: They tell you where the graph of the polynomial crosses or touches the <i>x</i><i>&#8211;</i>axis.</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":"2022-01-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207994},{"headers":{"creationTime":"2016-03-27T16:54:08+00:00","modifiedTime":"2022-03-14T14:35:38+00:00","timestamp":"2022-09-14T18:19:24+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Linear Algebra For Dummies Cheat Sheet","strippedTitle":"linear algebra for dummies cheat sheet","slug":"linear-algebra-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Must-know linear algebra concepts, including common numerical values of trig functions, basic algebraic properties and calculator commands.","noIndex":0,"noFollow":0},"content":"To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.","description":"To study and solve linear algebra equations successfully, you need to know common numerical values of trig functions, what elements determine a vector space, basic algebraic properties, and general commands using graphing calculators to solve linear algebra problems.","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":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":188400,"title":"Algebraic Properties You Should Know","slug":"algebraic-properties-you-should-know","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188400"}},{"articleId":188396,"title":"Commonly Used Values of Selected Trig Functions","slug":"commonly-used-values-of-selected-trig-functions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188396"}},{"articleId":188395,"title":"How to Meet Vector Space Requirements","slug":"how-to-meet-vector-space-requirements","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188395"}},{"articleId":188390,"title":"Calculator Commands for Linear Algebra","slug":"calculator-commands-for-linear-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188390"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282354,"slug":"linear-algebra-for-dummies","isbn":"9780470430903","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/0470430907/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/0470430907/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/0470430907-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/0470430907/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/0470430907/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/linear-algebra-for-dummies-cover-9780470430903-203x255.jpg","width":203,"height":255},"title":"Linear Algebra For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of numerous <i>For Dummies</i> books. She is a lecturer at Bradley University in Peoria, Illinois, where she has taught courses in algebra, calculus, and other mathematics topics for almost 30 years. </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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9780470430903&quot;]}]\" id=\"du-slot-63221b2c8accd\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9780470430903&quot;]}]\" id=\"du-slot-63221b2c8b6bb\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":188396,"title":"Commonly Used Values of Selected Trig Functions","slug":"commonly-used-values-of-selected-trig-functions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188396"}},{"articleId":188395,"title":"How to Meet Vector Space Requirements","slug":"how-to-meet-vector-space-requirements","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188395"}},{"articleId":188400,"title":"Algebraic Properties You Should Know","slug":"algebraic-properties-you-should-know","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188400"}},{"articleId":188390,"title":"Calculator Commands for Linear Algebra","slug":"calculator-commands-for-linear-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/188390"}}],"content":[{"title":"Commonly used values of selected trig functions","thumb":null,"image":null,"content":"<p>When performing transformations in trig functions, such as rotations, you need to use the numerical values of these functions. Here are some of the more commonly used angles.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/249344.image0.png\" alt=\"image0.png\" width=\"441\" height=\"347\" /></p>\n"},{"title":"How to meet vector space requirements","thumb":null,"image":null,"content":"<p>In linear algebra, a set of elements is termed a <em>vector space</em> when particular requirements are met. For example, let a set consist of vectors <strong>u</strong>, <strong>v</strong>, and <strong>w</strong>. Also let <em>k</em> and <em>l</em> be real numbers, and consider the defined operations of ⊕ and ⊗. The set is a vector space if, under the operation of ⊕, it meets the following requirements:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><strong>Closure. u</strong> ⊕ <strong>v</strong> is in the set.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Commutativity. u</strong> ⊕ <strong>v = v</strong> ⊕ <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Associativity. u</strong> ⊕ (<strong>v</strong> ⊕ <strong>w</strong>) = (<strong>u</strong> ⊕ <strong>v</strong>) ⊕ <strong>w.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>An identity element 0. u</strong> ⊕ <strong>0</strong> = <strong>0</strong> ⊕ <strong>u</strong> = <strong>u</strong> for any element <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>An inverse element −u. u</strong> ⊕ <strong>−u</strong> = <strong>−u</strong> ⊕<strong> u</strong> = <strong>0</strong></p>\n</li>\n</ul>\n<p>Under the operation of ⊗, the set is a vector space if it meets the following requirements:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><strong>Closure.</strong> <em>k</em> ⊗ <strong>u</strong> is in the set.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Distribution over a vector sum.</strong> <em>k</em> ⊗ (<strong>u</strong> ⊕ <strong>v</strong>) = <em>k</em> ⊗ <strong>u</strong> ⊕ <em>k</em>⊗ <strong>v</strong>.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Distribution over a scalar sum.</strong> (<em>k</em> + <em>l</em>) ⊗ <strong>u</strong> = <em>k</em> ⊗<strong>u</strong> ⊕ <em>l</em> ⊗ <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Associativity of a scalar product.</strong> <em>k</em> ⊗ (<em>l</em> ⊗ <strong>u</strong>) = (<em>kl</em>) ⊗ <strong>u.</strong></p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Multiplication by the scalar identity.</strong> 1 ⊗ <strong>u</strong> = <strong>u.</strong></p>\n</li>\n</ul>\n"},{"title":"Algebraic properties you should know","thumb":null,"image":null,"content":"<p>You can use a number of properties when working with linear algebraic expressions, including the commutative, associative, and distributive properties of addition and multiplication, as well as identities and inverses in addition and multiplication:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/249347.image0.png\" alt=\"image0.png\" width=\"420\" height=\"347\" /></p>\n"},{"title":"Calculator commands for linear algebra","thumb":null,"image":null,"content":"<p>Graphing calculators are wonderful tools for helping you solve linear algebra processes; they allow you to drain battery power rather than brain power. Since there is a wide variety of graphing calculators out there, the following are general instructions for help with linear algebra that apply to most graphing calculators:</p>\n<h2>To solve systems of equations by graphing:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Write each equation in <i>y</i> = <i>mx</i> + <i>b</i> form.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Insert equations in the <i>y</i>-menu.</p>\n</li>\n<li>\n<p class=\"first-para\">3. Graph the lines.</p>\n</li>\n<li>\n<p class=\"first-para\">4. Use the Intersection tool to get the answer.</p>\n</li>\n</ol>\n<h2>To add or subtract matrices:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into the matrices A and B.</p>\n</li>\n<li>\n<p class=\"first-para\">2. With a new screen, press [A] + [B] or [A] – [B], and press Enter.</p>\n</li>\n</ol>\n<h2>To multiply by a scalar:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into the matrix A.</p>\n</li>\n<li>\n<p class=\"first-para\">2. With a new screen, press the scalar and multiply: <i>k</i> * [A], and press Enter.</p>\n</li>\n</ol>\n<h2>To multiply two matrices together:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into the matrices A and B.</p>\n</li>\n<li>\n<p class=\"first-para\">2. With a new screen, press [A] * [B], and press Enter.</p>\n</li>\n</ol>\n<h2>To switch rows:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row swap</i>: <i>rowSwap</i> ([matrix name], first row, second row), and press Enter.</p>\n</li>\n</ol>\n<h2>To add two rows together:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row addition</i>: &#8220;<i>row</i> +&#8221;, ([matrix name], row to be added to target row, target row), and press Enter.</p>\n</li>\n</ol>\n<h2>To add the multiple of one row to another:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use row <i>sum</i>&#8211;<i>of</i>&#8211;<i>multiple</i>: &#8220;*<i>row</i> +&#8221;, (multiplier, [matrix name], row being multiplied, target row having multiple added to it), and press Enter.</p>\n</li>\n</ol>\n<h2>To multiply a row by a scalar:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row</i> <i>multiple</i>: &#8220;*row&#8221; (multiplier, [matrix name], row), and press Enter.</p>\n</li>\n</ol>\n<h2>To create an echelon form:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use <i>row</i>&#8211;<i>echelon</i> <i>form</i>: <i>ref</i> ([matrix name]) or <i>reduced row-echelon form: rref</i> ([matrix name]), and press Enter.</p>\n</li>\n</ol>\n<h2>To raise a matrix to a power:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use the caret operation with power, <i>p</i>: [matrix name] ^ <i>p</i>, and press Enter.</p>\n</li>\n</ol>\n<h2>To find inverses:</h2>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Insert the elements into a matrix.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Use the reciprocal operation, <i>x</i><sup>−1</sup>: [matrix name]<sup>−1</sup>, and press Enter.</p>\n</li>\n</ol>\n<h2>To solve systems of linear equations:</h2>\n<p>(This only works when the system has a single solution; it fails when the matrix A is singular.)</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">1. Write each equation with the variables in the same order and the constant on the other side of the equation sign.</p>\n</li>\n<li>\n<p class=\"first-para\">2. Create a matrix A, whose elements are the coefficients of the variables.</p>\n</li>\n<li>\n<p class=\"first-para\">3. Create a matrix B, whose elements are the constants.</p>\n</li>\n<li>\n<p class=\"first-para\">4. Press, A<sup>−1</sup> * B, and press Enter.</p>\n</li>\n</ol>\n<p>The resulting vector has the values of the variables, in order.</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":"2022-03-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":208656},{"headers":{"creationTime":"2016-03-27T16:57:12+00:00","modifiedTime":"2022-02-24T20:15:42+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra I For Dummies Cheat Sheet","strippedTitle":"algebra i for dummies cheat sheet","slug":"algebra-i-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Take the stress out of learning algebra by mastering the rules and formulas.","noIndex":0,"noFollow":0},"content":"Algebra problems are easier to solve when you know the rules and formulas. Memorizing key algebra formulas will speed up your work, too. And if you know the rules of divisibility and the order of operations, you'll be able to solve algebra problems without getting stressed.","description":"Algebra problems are easier to solve when you know the rules and formulas. Memorizing key algebra formulas will speed up your work, too. And if you know the rules of divisibility and the order of operations, you'll be able to solve algebra problems without getting stressed.","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":33721,"title":"Algebra","slug":"algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":193760,"title":"Algebra's Rules of Divisibility","slug":"algebras-rules-of-divisibility","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193760"}},{"articleId":193749,"title":"Algebra Formulas Worth Memorizing","slug":"algebra-formulas-worth-memorizing","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193749"}},{"articleId":193748,"title":"Order of Operations in Algebra","slug":"order-of-operations-in-algebra-2","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193748"}}],"fromCategory":[{"articleId":255800,"title":"Applying the Distributive Property: Algebra Practice Questions","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/255800"}},{"articleId":245778,"title":"Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210250"}},{"articleId":210249,"title":"Understanding Algebraic Variables","slug":"understanding-algebraic-variables","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/210249"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281938,"slug":"algebra-i-for-dummies-2nd-edition","isbn":"9781119293576","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/111929357X/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/111929357X-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/111929357X/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/algebra-i-for-dummies-2nd-edition-cover-9781119293576-202x255.jpg","width":202,"height":255},"title":"Algebra I For Dummies","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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293576&quot;]}]\" id=\"du-slot-63221b2258a19\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293576&quot;]}]\" id=\"du-slot-63221b22594db\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":193749,"title":"Algebra Formulas Worth Memorizing","slug":"algebra-formulas-worth-memorizing","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193749"}},{"articleId":193760,"title":"Algebra's Rules of Divisibility","slug":"algebras-rules-of-divisibility","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193760"}},{"articleId":193748,"title":"Order of Operations in Algebra","slug":"order-of-operations-in-algebra-2","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/193748"}}],"content":[{"title":"Algebra formulas worth memorizing ","thumb":null,"image":null,"content":"<div id=\"article-content-wrapper\" data-v-89b5964a=\"\" data-v-76ccd6ad=\"\">\n<article class=\"article-content w-100\" data-v-89b5964a=\"\" data-testid=\"articleContent\">Some formulas occur frequently when you&#8217;re doing algebraic manipulations and working through mathematical applications. You&#8217;ll find ways to use these algebra formulas even when you&#8217;re doing something other than algebra, such as planning a garden or a road trip. Of course, you need to know what the letters and symbols in the formula mean, so both the formulas and the explanations are in this table.</p>\n<div class=\"img-div\"><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/175704.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"567\" /></div>\n</article>\n</div>\n<div class=\"about-article-section\" data-v-2c35db38=\"\" data-v-76ccd6ad=\"\">\n<h2 class=\"h3 section-title\" data-v-2c35db38=\"\" data-testid=\"aboutArticleHead\"></h2>\n</div>\n"},{"title":"Algebra's rules of divisibility ","thumb":null,"image":null,"content":"<p>In algebra, knowing the rules of divisibility can help you solve faster. When factoring algebraic expressions to solve equations, you need to be able to pull out the greatest factor. You also need common factors when reducing algebraic fractions. The rules of divisibility help you find the common factors and change the algebraic expressions so that they&#8217;re put in a more workable form.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Divisibility by 2:</b> A number is divisible by 2 if the last digit in the number is 0, 2, 4, 6, or 8.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 3:</b> A number is divisible by 3 if the sum of the digits in the number is divisible by 3.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 4:</b> A number is divisible by 4 if the last two digits in the number form a number divisible by 4.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 5:</b> A number is divisible by 5 if the last digit is 0 or 5.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 6:</b> A number is divisible by 6 if it is divisible by both 2 and 3.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 8:</b> A number is divisible by 8 if the last three digits form a number divisible by 8.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 9:</b> A number is divisible by 9 if the sum of the digits of the number is divisible by 9.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 10:</b> A number is divisible by 10 if it ends in 0.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 11:</b> A number is divisible by 11 if the sums of the alternate digits are different by 0, 11, 22, or 33, or any two-digit multiple of 11. In other words, say you have a six-digit number: Add up the first, third, and fifth digits — the odd ones. Then add the digits in the even places — second, fourth, and sixth. Then subtract the smaller of those totals from the larger total, and if the answer is a multiple of 11, the original number is divisible by 11.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Divisibility by 12: </b>A number is divisible by 12 if the last two digits form a number divisible by 4 and if the sum of the digits is divisible by 3.</p>\n</li>\n</ul>\n"},{"title":"Order of operations in algebra ","thumb":null,"image":null,"content":"<p>Solve algebra problems correctly by following the order of operations. When performing more than one operation on an algebraic expression, work out the operations and signs in the following order:</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">First calculate powers and roots.</p>\n</li>\n<li>\n<p class=\"first-para\">Then perform all multiplication and division.</p>\n</li>\n<li>\n<p class=\"first-para\">Finish with addition and subtraction.</p>\n</li>\n</ol>\n<p>If you have more than two operations of the same level, do them in order from left to right, following the order of operations.</p>\n<p>For example, to solve 24 ÷ 3 + 11 – 3<sup>2</sup> × 2, you would</p>\n<ol class=\"level-one\">\n<li>\n<p class=\"first-para\">First calculate powers and roots.</p>\n<p class=\"child-para\">This problem doesn&#8217;t have any roots, but it does have one power, 3<sup>2</sup>. You know that 3<sup>2</sup> = 3 × 3 = 9. Substitute 9 into the problem, and you get 24 ÷ 3 + 11 – 9 × 2.</p>\n</li>\n<li>\n<p class=\"first-para\">Then perform all multiplication and division.</p>\n<p class=\"child-para\">Working left to right, 24 ÷ 3 = 8 and 9 × 2 = 18. Substitute those numbers into the problem, and you have 8 + 11 – 18.</p>\n</li>\n<li>\n<p class=\"first-para\">Finish with addition and subtraction.</p>\n<p class=\"child-para\">So, you end up with 8 + 11 – 18 = 19 – 18 = 1.</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":"Two years","lifeExpectancySetFrom":"2022-02-24T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":209219},{"headers":{"creationTime":"2016-03-27T16:47:10+00:00","modifiedTime":"2022-01-21T18:01:48+00:00","timestamp":"2022-09-14T18:19:02+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":"Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Algebra II For Dummies Cheat Sheet","strippedTitle":"algebra ii for dummies cheat sheet","slug":"algebra-ii-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This handy guide includes the algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and more.","noIndex":0,"noFollow":0},"content":"Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!","description":"Algebra is all about formulas, equations, and graphs. You need algebraic equations for multiplying binomials, dealing with radicals, finding the sum of sequences, and graphing the intersections of cones and planes. You also get to deal with logarithms, you lucky Algebra II user!","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. 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You multiply the sum and difference of binomials and multiply by squaring and cubing to find some of the special products in algebra. See if you can spot the patterns in these equations:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Sum and difference: (<em>a</em> + <em>b</em>)(<em>a</em> – <em>b</em>) = <em>a</em><sup>2</sup> – <em>b</em><sup>2</sup></p>\n</li>\n<li>\n<p class=\"first-para\">Binomial squared: (<em>a</em> + <em>b</em>)<sup>2</sup> = <em>a</em><sup>2</sup> + 2<em>ab</em> + <em>b</em><sup>2</sup></p>\n</li>\n<li>\n<p class=\"first-para\">Binomial cubed: (<em>a</em> + <em>b</em>)<sup>3</sup> = <em>a</em><sup>3</sup> + 3<em>a</em><sup>2</sup><em>b </em>+ 3<em>ab</em><sup>2</sup> + <em>b</em><sup>3</sup></p>\n</li>\n</ul>\n"},{"title":"Algebra's quadratic formula","thumb":null,"image":null,"content":"<p>You can find solutions for quadratic equations by factoring, completing the square, guessing, or everyone&#8217;s favorite — using the quadratic formula. The best thing about the quadratic formula (unlike the other methods) is that it always works.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132935.image0.jpg\" alt=\"image0.jpg\" width=\"404\" height=\"134\" /></p>\n"},{"title":"Rules for algebraic radicals","thumb":null,"image":null,"content":"<p>Working with radicals can be troublesome, but these equivalences keep algebraic radicals from running amok. Finding the root of product or quotient or a fractional exponent is simple with these formulas; just be sure that the numbers replacing the factors <i>a</i> and <i>b</i> are positive.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132937.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"213\" /></p>\n"},{"title":"Algebra’s laws of logarithms","thumb":null,"image":null,"content":"<p>Logarithms help you add instead of multiply. The algebra formulas here make it easy to find equivalence, the logarithm of a product, quotient, power, reciprocal, base, and the log of 1.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132939.image0.jpg\" alt=\"image0.jpg\" width=\"260\" height=\"400\" /></p>\n"},{"title":"Standard equations of algebraic conics","thumb":null,"image":null,"content":"<p>Conics are curved algebraic forms that come from slicing a cone with a plane. Use these equations to graph algebraic conics, such as circles, ellipses, parabolas, and hyperbolas:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/477053.image0.jpg\" alt=\"image0.jpg\" width=\"499\" height=\"400\" /></p>\n"},{"title":"Cramer's rule for linear algebra","thumb":null,"image":null,"content":"<p>Named for Gabriel Cramer, Cramer’s Rule provides a solution for a system of two linear algebraic equations in terms of determinants — the numbers associated with a specific, square matrix.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132926.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"49\" /></p>\n"},{"title":"Using algebra to find the sums of sequences","thumb":null,"image":null,"content":"<p>Algebra can help you add a series of numbers (the sum of sequences) more quickly than you would be able to with straight addition. Adding integers, squares, cubes, and terms in an arithmetic or geometric sequence is simple with these algebraic formulas:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/254965.image0.png\" alt=\"image0.png\" width=\"447\" height=\"436\" /></p>\n"},{"title":"Algebraic Permutations and Combinations","thumb":null,"image":null,"content":"<p>In algebra, you use permutations to count the number of subsets of a larger set. Use permutations when order is necessary. With combinations, you can count the number of subsets when order doesn&#8217;t matter. The formulas you need are</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132930.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"209\" /></p>\n"},{"title":"Eight basic algebraic curves","thumb":null,"image":null,"content":"<p>Algebra is all about graphing relationships, and the curve is one of the most basic shapes used. Here&#8217;s a look at eight of the most frequently used graphs.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/132932.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"886\" /></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 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Algebra Articles

Master algebra and algebra II with quick breakdowns of key math skills, plus practice problem sets and other cool stuff.

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Algebra Algebra II All-in-One For Dummies Cheat Sheet

Cheat Sheet / Updated 10-10-2022

Here it is. You have this All-in-One reference for concepts and formulas occurring in Algebra II. The material here is grouped by general algebraic content to make it easier to find what you need. The formulas have the standard mathematical format with variables appearing as x, y, and z and the constant numbers appearing as letters at the beginning of the alphabet.

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Algebra Algebra II: Making Matrices Work for You

Article / Updated 09-22-2022

A matrix is a rectangular array of numbers. Each row has the same number of elements, and each column has the same number of elements. Matrices can be classified as: square, identity, zero, column, and so on. Where did matrices come from? For most of their history, they were called arrays. There are references to arrays in Chinese, French, Italian, and many other mathematical works going back many hundreds of years. American mathematician George Dantzig's work with matrices during World War II allowed for the coordination of shipments of supplies and troops to various locations. Matrices are here to stay. You may be familiar with a method used to solve systems of linear equations using matrices, but this application just scratches the surface of what matrices can do. First, just in case you're not familiar with solving equations using matrices, let me give just a quick description. If you want to solve the following system of equations: You write the matrix: And then you perform row operations until you get the matrix: From that matrix, you know that the solution of the system of equations is x = 1, y = -3, and z = -5. Pretty slick, don't you think? But uses for matrices don't stop there. You can solve traffic control problems, transportation logistics problems (how much of each item to send to various distribution centers), dietary problems (how much of each food product is needed to meet several different dietary requirements), and so on. Matrices work well in graphing calculators and computer spreadsheets — just set up the problem and let the technology do all the work.

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Algebra How to Write Numbers in Scientific Notation

Article / Updated 08-17-2022

Scientific notation is a standard way of writing very large and very small numbers so that they're easier to both compare and use in computations. To write in scientific notation, follow the form where N is a number between 1 and 10, but not 10 itself, and a is an integer (positive or negative number). You move the decimal point of a number until the new form is a number from 1 up to 10 (N), and then record the exponent (a) as the number of places the decimal point was moved. Whether the power of 10 is positive or negative depends on whether you move the decimal to the right or to the left. Moving the decimal to the right makes the exponent negative; moving it to the left gives you a positive exponent. To see an exponent that's positive, write 312,000,000,000 in scientific notation: Move the decimal place to the left to create a new number from 1 up to 10. Where's the decimal point in 312,000,000,000? Because it's a whole number, the decimal point is understood to be at the end of the number: 312,000,000,000. So, N = 3.12. Determine the exponent, which is the number of times you moved the decimal. In this example, you moved the decimal 11 times; also, because you moved the decimal to the left, the exponent is positive. Therefore, a = 11, and so you get Put the number in the correct form for scientific notation To see an exponent that's negative, write .00000031 in scientific notation. Move the decimal place to the right to create a new number from 1 up to 10. So, N = 3.1. Determine the exponent, which is the number of times you moved the decimal. In this example, you moved the decimal 7 times; also, because you moved the decimal to the right, the exponent is negative. Therefore, a = –7, and so you get Put the number in the correct form for scientific notation When you get used to writing numbers in scientific notation, you can do it all in one step. Here are a few examples: Order of magnitude Why does scientific notation always use a decimal between 1 and 10? The answer has to do with order of magnitude, which is a simple way to keep track of roughly how large a number is so you can compare numbers more easily. The order of magnitude of a number is its exponent in scientific notation. For example, 703 = 7.03 x 102 — order of magnitude is 2 600,000 = 6 x 105 — order of magnitude is 5 0.00095 = 9.5 x 10–4 — order of magnitude is –4 Every number between 10 and 100 has an order of magnitude of 1. Every number between 100 and 1,000 has an order of magnitude of 2.

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