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class=\"Tip\">Note that ln(<i>ax</i>) differentiates to <i>a</i> / <i>ax</i>, and after you cancel the <i>a</i>s you’re left with 1/<i>x</i>.</p>\n"},{"title":"Differentiating combinations","thumb":null,"image":null,"content":"<p>Once you’ve mastered the building blocks of differentiation for A-level maths, the next step is to get comfortable with the various rules – what happens when you have a function with something complicated inside the brackets? How about if you have expressions involving addition, subtraction, multiplication or division? This table reminds you what to do in these situations, including the dreaded chain rule, quotient rule and product rule from Core 3.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/503355.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"168\" /></p>\n<p class=\"Remember\">If you’re doing <i>p</i><i>arametric differentiation</i><i> </i>in Core 4, then <i>d</i><i>y</i>/<i>d</i><i>x</i> = (<i>d</i><i>y</i>/<i>d</i><i>t</i>)/(<i>d</i><i>x</i>/<i>d</i><i>t</i>).</p>\n"},{"title":"Integrating simple terms and functions","thumb":null,"image":null,"content":"<p>If you want to get a good grade in A-level maths, one of the skills you need is the integration of simple terms – powers of <i>x</i>, exponentials and trigonometric functions.</p>\n<p class=\"Remember\">Any time you integrate, you have to remember to add a constant! 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But trigonometry also has some special formulas usually found just in those discussions.</p>\n<p>A formula provides you a rule or equation that you can count on to work, every single time. A formula gives a relationship between particular quantities and units. The main trick to using formulas is to know what the different letters represent.</p>\n<p>In the formulas given here, you have: <i>r</i> (radius); <i>d</i> (diameter or distance); <i>b</i> (base or measure of a side); <i>h</i> (height); <i>a</i>, <i>b</i>, <i>c</i> (measures of sides); <i>x</i>, <i>y</i> (coordinates on a graph); <i>m</i> (slope); <i>M</i> (midpoint); <i>h</i>, <i>k</i> (horizontal and vertical distances from the center); <i>θ</i> (angle theta); and <i>s</i> (arc length).</p>\n<p>The formulas particular to trigonometry have: sin (sine), cos (cosine), and tan (tangent), although only <i>sin</i> is represented here.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/413261.image0.jpg\" alt=\"image0.jpg\" width=\"285\" height=\"400\" /></p>\n"},{"title":"Special right triangles","thumb":null,"image":null,"content":"<p>Every right triangle has the property that the sum of the squares of the two legs is equal to the square of the <i>hypotenuse</i> (the longest side). The Pythagorean theorem is written: <i>a</i><sup>2</sup> + <i>b</i><sup>2</sup> = <i>c</i><sup>2</sup>. What’s so special about the two right triangles shown here is that you have an even more special relationship between the measures of the sides — one that goes beyond (but still works with) the Pythagorean theorem.</p>\n<p>When you have a 30-60-90 right triangle, the measure of the hypotenuse is always twice the measure of the shortest side, and the other leg is always</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/411654.image0.png\" alt=\"image0.png\" width=\"28\" height=\"28\" /></p>\n<p>or about 1.7 times as big as the shortest side. With the isosceles right triangle, the two legs measure the same, and the hypotenuse is always</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/411655.image1.png\" alt=\"image1.png\" width=\"28\" height=\"28\" /></p>\n<p>or about 1.4 times as long as those two legs.</p>\n<p>&nbsp;</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-297181\" src=\"https://www.dummies.com/wp-content/uploads/9781394168552-cs02.jpg\" alt=\"\" width=\"535\" height=\"344\" /></p>\n"},{"title":"Right triangle definitions for trigonometry functions","thumb":null,"image":null,"content":"<p>The basic trig functions can be defined with ratios created by dividing the lengths of the sides of a right triangle in a specific order. The label <i>hypotenuse</i> always remains the same — it’s the longest side. But the designations of <i>opposite</i> and <i>adjacent</i> can change — depending on which angle you’re referring to at the time. The <i>opposite</i> side is always that side that doesn’t help make up the angle, and the <i>adjacent</i> side is always one of the sides of the angle.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/411616.image0.jpg\" alt=\"image0.jpg\" width=\"387\" height=\"400\" /></p>\n"},{"title":"Coordinate definitions for trigonometry functions","thumb":null,"image":null,"content":"<p>The trig functions can be defined using the measures of the sides of a right triangle. But they also have very useful definitions using the coordinates of points on a graph.</p>\n<p>First, let let the vertex of an angle be at the origin — the point (0,0) — and let the initial side of that angle lie along the positive <i>x</i>-axis and the terminal side be a rotation in a counterclockwise motion. Then, when the point (<i>x</i>,<i>y</i>) lies on a circle that’s intersected by that terminal side, the trig functions are defined with the following ratios, where <i>r</i> is the radius of the circle.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/411618.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"172\" /></p>\n"},{"title":"Signs of trigonometry functions in quadrants","thumb":null,"image":null,"content":"<p>An angle is in <i>standard position</i> when its vertex is at the origin, its initial side is on the positive <i>x</i>-axis, and the terminal side rotates counterclockwise from the initial side. The position of the terminal side determines the sign of the various trig functions of that angle. The following shows you which functions are positive — and you can assume that the other functions are negative in that quadrant.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/411620.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"303\" /></p>\n"},{"title":"Degree/radian equivalences for selected angles","thumb":null,"image":null,"content":"<p>As you study trigonometry, you&#8217;ll find occasions when you need to change degrees to radians, or vice versa. A formula for changing from degrees to radians or radians to degrees is:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-297182\" src=\"https://www.dummies.com/wp-content/uploads/9781394168552-cs06.jpg\" alt=\"\" width=\"150\" height=\"104\" /></p>\n<p>&nbsp;</p>\n<p>The formula works for any angle, but the most commonly used angles and their equivalences are shown below.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/413264.image1.jpg\" alt=\"image1.jpg\" width=\"535\" height=\"57\" /></p>\n"},{"title":"Laws of sines and cosines","thumb":null,"image":null,"content":"<p>The laws of sines and cosines give you relationships between the lengths of the sides and the trig functions of the angles. These laws are used when you don’t have a right triangle — they work in any triangle. You determine which law to use based on what information you have. In general, the side <i>a</i> lies opposite angle <i>A</i>, the side <i>b</i> is opposite angle <i>B</i>, and side <i>c </i>is opposite angle <i>C</i>.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/411625.image0.jpg\" alt=\"image0.jpg\" width=\"276\" height=\"400\" /></p>\n"},{"title":"Exact trigonometry functions for selected acute angles","thumb":null,"image":null,"content":"<p>Using the lengths of the sides of the two special right triangles — the 30-60-90 right triangle and the 45-45-90 right triangle — the following exact values for trig functions are found.</p>\n<p>Using these values in conjunction with reference angles and signs of the functions in the different quadrants, you can determine the exact values of the multiples of these angles.</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/411627.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"368\" /></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":"2023-02-09T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207754},{"headers":{"creationTime":"2022-12-27T19:34:11+00:00","modifiedTime":"2023-01-19T16:12:23+00:00","timestamp":"2023-01-19T18:01:02+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":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"Statistics All-in-One For Dummies Cheat Sheet","strippedTitle":"statistics all-in-one for dummies cheat sheet","slug":"statistics-all-in-one-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This Cheat Sheet is a handy reference to help you organize, understand, and remember the notation and formulas for statistics.","noIndex":0,"noFollow":0},"content":"Statistics involves a lot of data analysis, and analysis is built with math notation and formulas — but never fear, your cheat sheet is here to help you organize, understand, and remember the notation and formulas so that when it comes to putting them into practice or to the test, you’re ahead of the game!","description":"Statistics involves a lot of data analysis, and analysis is built with math notation and formulas — but never fear, your cheat sheet is here to help you organize, understand, and remember the notation and formulas so that when it comes to putting them into practice or to the test, you’re ahead of the game!","blurb":"","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":" <p><b>Deborah Rumsey, PhD, </b>is a Professor of Statistics and Statistics Education Specialist at The Ohio State University. She is the author of <i>Statistics For Dummies, Statistics Workbook For Dummies, Statistics II For Dummies, </i>and<i> Probability For Dummies</i>. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"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":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat Sheet","slug":"spss-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209293"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":295383,"slug":"statistics-all-in-one-for-dummies","isbn":"9781119902560","categoryList":["academics-the-arts","math","statistics"],"amazon":{"default":"https://www.amazon.com/gp/product/1119902568/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119902568/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/1119902568-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119902568/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119902568/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/statistics-all-in-one-for-dummies-cover-9781119902560-204x255.jpg","width":204,"height":255},"title":"Statistics All-in-One For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b>Deborah Rumsey, PhD, </b>is a Professor of Statistics and Statistics Education Specialist at The Ohio State University. She is the author of <i>Statistics For Dummies, Statistics Workbook For Dummies, Statistics II For Dummies, </i>and<i> Probability For Dummies</i>.</p>","authors":[{"authorId":9121,"name":"Deborah J. Rumsey","slug":"deborah-j-rumsey","description":" <p><b>Deborah Rumsey, PhD, </b>is a Professor of Statistics and Statistics Education Specialist at The Ohio State University. She is the author of <i>Statistics For Dummies, Statistics Workbook For Dummies, Statistics II For Dummies, </i>and<i> Probability For Dummies</i>. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9121"}}],"_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;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119902560&quot;]}]\" id=\"du-slot-63c9855ec46d7\"></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;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119902560&quot;]}]\" id=\"du-slot-63c9855ec56fc\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":0,"title":"","slug":null,"categoryList":[],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/"}}],"content":[{"title":"10 tips for the statistically savvy sleuth","thumb":null,"image":null,"content":"<p>Here are some tips on what to do when you encounter statistics in the real world or in the workplace so you can avoid some of the most common statistical mistakes out there. Don’t just read and believe!</p>\n<ul>\n<li>Pinpoint misleading graphs</li>\n<li>Uncover biased data</li>\n<li>Search for a margin of error</li>\n<li>Identify nonrandom samples</li>\n<li>Notice missing sample sizes</li>\n<li> Detect misinterpreted correlations</li>\n<li>Point out confounding variables</li>\n<li>Inspect the numbers</li>\n<li>Report selective reporting</li>\n<li>Expose the anecdote</li>\n</ul>\n"},{"title":"10 surefire exam score boosters","thumb":null,"image":null,"content":"<p>Want to get a higher score on your next exam? Put these ten tips into practice to get ahead in your class — and stay ahead.</p>\n<ul>\n<li>Figure out what you don’t know, then do something about it.</li>\n<li>Avoid saying, “Yeah, yeah I know this,” when you, in fact, may not — be active, not passive.</li>\n<li>Make friends with formulas.</li>\n<li>Make an “If-Then” chart. IF my professor asks me <em>X</em>, THEN here’s what I do: <em>Y</em>.</li>\n<li>Figure out exactly what the question is asking.</li>\n<li>Label what you are given in the problem.</li>\n<li>Draw a picture.</li>\n<li>Connect what’s given to what’s being asked, and you can solve the problem.</li>\n<li>Do the math — twice — just to make sure.</li>\n<li>Analyze your answer: Does it make sense?</li>\n</ul>\n"},{"title":"Notation and meanings","thumb":null,"image":null,"content":"<p>Notation is math shorthand for the ideas and concepts used to represent sample and population values. Following are the most common symbols in statistics and what they stand for. Keep this list handy so that you don’t get tripped up by notation.</p>\n<h3>Values and statistics from a sample</h3>\n<p><em>x̄</em> is the mean of one sample of quantitative data.</p>\n<p><em>s</em> is the standard deviation of one sample of quantitative data.</p>\n<p><em>p̂</em> is the proportion of yeses (successes) in one sample of a binomial data set.</p>\n<h3>Values and parameters from a population</h3>\n<p><em>X</em> is a random variable that represents any individual value in the population.</p>\n<p><em>x </em>is the value of <em>X</em> for one specific individual from the population.</p>\n<p><em>μ<sub>x</sub></em> is the mean of the random variable <em>X</em> — also known as the population mean.</p>\n<p><em>σ<sub>x</sub></em> is the standard deviation of <em>X</em> — also known as the population standard deviation.</p>\n<p><em>p </em>is the proportion of yeses (successes) in the entire binomial population.</p>\n<h3>Parameters and statistics regarding the sample mean</h3>\n<p><em>X̄</em> is a random variable that represents any sample mean of any sample of size <em>n</em> taken from the population.</p>\n<p><em>μ<sub>x̄</sub></em> is the mean of <em>X̄</em> — also known as the mean of all possible values of <em>μ<sub>x̄</sub></em>. Note that <em>μ<sub>x̄</sub></em> = <em>μ<sub>x</sub></em>.</p>\n<p><em>σ<sub>x̄</sub></em> is the standard error of <em>X̄</em> — also known as the standard deviation of all possible values of <em>X̄</em>. Note that <em>σ<sub>x̄</sub></em> = <em>σ<sub>x</sub>/√n</em>.</p>\n"},{"title":"Distributions","thumb":null,"image":null,"content":"<p>Three distributions are very common in statistics: normal and binomial distributions and sampling distributions. Following are the formulas that are used with each of these three distributions to help solve problems.</p>\n<p>Normal: Use<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296526\" src=\"https://www.dummies.com/wp-content/uploads/normal-distribution-formula.png\" alt=\"normal distribution formula\" width=\"87\" height=\"47\" /></p>\n<p>Binomial: Mean <em>np</em>; Variance <em>np</em>(1 &#8211; <em>p</em>).</p>\n<blockquote><p>Probability:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296531\" src=\"https://www.dummies.com/wp-content/uploads/binomial-distribution-probability-formula.png\" alt=\"binomial distribution probability formula\" width=\"335\" height=\"60\" /></p></blockquote>\n<blockquote><p>Normal Approximation:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296529\" src=\"https://www.dummies.com/wp-content/uploads/binomial-distribution-normal-approximation-formula.png\" alt=\"binomial distribution normal approximation formula\" width=\"377\" height=\"56\" /></p></blockquote>\n<p>Sampling Distribution of <em>X̄</em>:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296527\" src=\"https://www.dummies.com/wp-content/uploads/sampling-distribution.png\" alt=\"sampling distribution formula\" width=\"356\" height=\"73\" /></p>\n"},{"title":"Confidence intervals ","thumb":null,"image":null,"content":"<p>Many confidence intervals exist, with the primary purpose of estimating a population parameter. The following list provides a host of formulas for many different confidence interval scenarios from one-sample confidence intervals to those involving two samples from two populations.</p>\n<p>One population mean <em>μ<sub>x</sub></em> with standard deviation known:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296534\" src=\"https://www.dummies.com/wp-content/uploads/confidence-interval-standard-deviation-known.png\" alt=\"confidence interval for one population mean &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; with standard deviation known\" width=\"360\" height=\"46\" /></p>\n<p>One population mean <em>μ<sub>x</sub></em> with standard deviation unknown:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296535\" src=\"https://www.dummies.com/wp-content/uploads/confidence-interval-standard-deviation-unknown.png\" alt=\"confidence interval for one population mean &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; with standard deviation unknown\" width=\"117\" height=\"42\" /></p>\n<p>One population proportion <em>p</em> with standard deviation unknown:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296533\" src=\"https://www.dummies.com/wp-content/uploads/confidence-interval-p-standard-deviation-unknown.png\" alt=\"confidence interval for one population proportion &lt;em&gt;p&lt;/em&gt; with standard deviation unknown\" width=\"365\" height=\"55\" /></p>\n<p>Difference of two population means <em>μ<sub>x</sub></em> &#8211; <em>μ<sub>y</sub></em> with standard deviations known:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296536\" src=\"https://www.dummies.com/wp-content/uploads/diff-two-population-means-sd-known.png\" alt=\"confidence interval for difference of two population means &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; - &lt;em&gt;μ&lt;sub&gt;y&lt;/sub&gt;&lt;/em&gt; with standard deviations known\" width=\"520\" height=\"65\" /></p>\n<p>Difference of two population means <em>μ<sub>x</sub></em> &#8211; <em>μ<sub>y</sub></em> with standard deviations unknown but equal:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296537\" src=\"https://www.dummies.com/wp-content/uploads/diff-two-population-means-sd-unknown-equal.png\" alt=\"confidence interval for difference of two population means &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; - &lt;em&gt;μ&lt;sub&gt;y&lt;/sub&gt;&lt;/em&gt; with standard deviations unknown but equal\" width=\"486\" height=\"65\" /></p>\n<p>Difference of two population means <em>μ<sub>x</sub></em> &#8211; <em>μ<sub>y</sub></em> with standard deviations unknown but unequal:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296538\" src=\"https://www.dummies.com/wp-content/uploads/diff-two-population-means-sd-unknown-unequal.png\" alt=\"confidence interval for difference of two population means &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; - &lt;em&gt;μ&lt;sub&gt;y&lt;/sub&gt;&lt;/em&gt; with standard deviations unknown but unequal\" width=\"398\" height=\"150\" /></p>\n<p>Difference of two population proportions <em>p<sub>1</sub></em> &#8211; <em>p<sub>2</sub></em> with standard deviations unknown:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296539\" src=\"https://www.dummies.com/wp-content/uploads/diff-two-population-proportions-sd-unknown.png\" alt=\"confidence internval for difference of two population proportions &lt;em&gt;p&lt;sub&gt;1&lt;/sub&gt;&lt;/em&gt; - &lt;em&gt;p&lt;sub&gt;2&lt;/sub&gt;&lt;/em&gt; with standard deviations unknown\" width=\"651\" height=\"67\" /></p>\n"},{"title":"Test statistics for hypothesis tests","thumb":null,"image":null,"content":"<p>Many hypothesis tests exist, with the primary purpose of testing a population parameter. Following are all the test statistics for both one-sample and two-sample hypothesis tests.</p>\n<p>One population mean <em>μ<sub>x</sub></em> with standard deviation known:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296522\" src=\"https://www.dummies.com/wp-content/uploads/hypothesis-test-one-pop-mean-sd-known.png\" alt=\"hypothesis test for one population mean &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; with standard deviation known\" width=\"300\" height=\"74\" /></p>\n<p>One population mean <em>μ<sub>x</sub></em> with standard deviation unknown:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296523\" src=\"https://www.dummies.com/wp-content/uploads/hypothesis-test-one-pop-mean-sd-unknown.png\" alt=\"hypothesis test for one population mean &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; with standard deviation unknown\" width=\"384\" height=\"67\" /></p>\n<p>One population proportion <em>p</em>:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296524\" src=\"https://www.dummies.com/wp-content/uploads/hypothesis-test-one-pop-proportion-p.png\" alt=\"hypothesis test for one population proportion &lt;em&gt;p&lt;/em&gt;\" width=\"357\" height=\"81\" /></p>\n<p>Difference of two population means <em>μ<sub>x</sub></em> &#8211; <em>μ<sub>y</sub></em> with standard deviations known:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296540\" src=\"https://www.dummies.com/wp-content/uploads/hypothesis-test-diff-two-pop-means-sd-known.png\" alt=\"hypothesis test for difference of two population means &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; - &lt;em&gt;μ&lt;sub&gt;y&lt;/sub&gt;&lt;/em&gt; with standard deviations known\" width=\"498\" height=\"92\" /></p>\n<p>Difference of two population means <em>μ<sub>x</sub></em> &#8211; <em>μ<sub>y</sub></em> with standard deviations unknown but equal:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296541\" src=\"https://www.dummies.com/wp-content/uploads/hypothesis-test-diff-two-pop-means-sd-unknown-equal.png\" alt=\"hypothesis test for difference of two population means &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; - &lt;em&gt;μ&lt;sub&gt;y&lt;/sub&gt;&lt;/em&gt; with standard deviations unknown but equal\" width=\"458\" height=\"96\" /></p>\n<p>Difference of two population means <em>μ<sub>x</sub></em> &#8211; <em>μ<sub>y</sub></em> with standard deviations unknown but unequal:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296542\" src=\"https://www.dummies.com/wp-content/uploads/hypothesis-test-diff-two-pop-means-sd-unknown-unequal.png\" alt=\"hypothesis test for difference of two population means &lt;em&gt;μ&lt;sub&gt;x&lt;/sub&gt;&lt;/em&gt; - &lt;em&gt;μ&lt;sub&gt;y&lt;/sub&gt;&lt;/em&gt; with standard deviations unknown but unequal\" width=\"370\" height=\"156\" /></p>\n<p>Two population proportions:<br />\n<img loading=\"lazy\" class=\"aligncenter size-full wp-image-296525\" src=\"https://www.dummies.com/wp-content/uploads/hypothesis-test-two-population-proporations.png\" alt=\"hypothesis test for two population proportions\" width=\"369\" height=\"96\" /></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":"2022-10-25T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":295421},{"headers":{"creationTime":"2016-03-26T21:47:50+00:00","modifiedTime":"2022-12-21T19:38:51+00:00","timestamp":"2022-12-21T21:01:02+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: How to Multiply and Divide Exponents","strippedTitle":"algebra: how to multiply and divide exponents","slug":"how-to-divide-exponents","canonicalUrl":"","seo":{"metaDescription":"Exponents show up in a variety of different math formats, equations, and formulas. Here's how you can multiply and divide them with ease.","noIndex":0,"noFollow":0},"content":"So, what is an exponent anyway? According to the Oxford dictionary, an <em>exponent</em> is defined as \"a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression.\" Exponents are used in almost all levels of math, from algebra to calculus to physics. Here are two ways you can work with exponents when they show up in formulas and equations.\r\n<h2 id=\"tab1\" >How to multiply exponents</h2>\r\nYou can multiply many exponential expressions together without having to change their form into the big or small numbers they represent. When multiplying exponents, the only requirement is that the bases of the exponential expressions have to be the same. So, you can multiply\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165070.image0.png\" alt=\"image0.png\" width=\"454\" height=\"38\" />\r\n\r\nbecause the bases are not the same (although the exponents are).\r\n<p class=\"Remember\">To multiply powers of the same base, add the exponents together:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165071.image1.png\" alt=\"image1.png\" width=\"188\" height=\"106\" />\r\n\r\nIf there’s more than one base in an expression with powers, you can combine the numbers with the same bases, find the values, and then write them all together. For example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165072.image2.png\" alt=\"image2.png\" width=\"368\" height=\"32\" />\r\n\r\nHere's an example with a number that has no exponent showing:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165073.image3.png\" alt=\"image3.png\" width=\"304\" height=\"38\" />\r\n<p class=\"Tip\">When there’s no exponent showing, such as with <i>y</i><i>,</i> you assume that the exponent is 1, so in the above example, you write</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165074.image4.png\" alt=\"image4.png\" width=\"26\" height=\"38\" />\r\n<h2 id=\"tab2\" >How to divide exponents</h2>\r\nYou can divide exponential expressions, leaving the answers as exponential expressions, as long as the bases are the same. To divide exponents (or powers) with the same base, subtract the exponents. Division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.\r\n\r\nFor example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165077.image0.png\" alt=\"image0.png\" width=\"194\" height=\"32\" />\r\n\r\nPretty easy, huh? Now wrap your brain around this:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165078.image1.png\" alt=\"image1.png\" width=\"406\" height=\"74\" />\r\n<p class=\"Remember\">Any number to the power of zero equals 1, as long as the base number is not 0.</p>","description":"So, what is an exponent anyway? According to the Oxford dictionary, an <em>exponent</em> is defined as \"a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression.\" Exponents are used in almost all levels of math, from algebra to calculus to physics. Here are two ways you can work with exponents when they show up in formulas and equations.\r\n<h2 id=\"tab1\" >How to multiply exponents</h2>\r\nYou can multiply many exponential expressions together without having to change their form into the big or small numbers they represent. When multiplying exponents, the only requirement is that the bases of the exponential expressions have to be the same. So, you can multiply\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165070.image0.png\" alt=\"image0.png\" width=\"454\" height=\"38\" />\r\n\r\nbecause the bases are not the same (although the exponents are).\r\n<p class=\"Remember\">To multiply powers of the same base, add the exponents together:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165071.image1.png\" alt=\"image1.png\" width=\"188\" height=\"106\" />\r\n\r\nIf there’s more than one base in an expression with powers, you can combine the numbers with the same bases, find the values, and then write them all together. For example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165072.image2.png\" alt=\"image2.png\" width=\"368\" height=\"32\" />\r\n\r\nHere's an example with a number that has no exponent showing:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165073.image3.png\" alt=\"image3.png\" width=\"304\" height=\"38\" />\r\n<p class=\"Tip\">When there’s no exponent showing, such as with <i>y</i><i>,</i> you assume that the exponent is 1, so in the above example, you write</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165074.image4.png\" alt=\"image4.png\" width=\"26\" height=\"38\" />\r\n<h2 id=\"tab2\" >How to divide exponents</h2>\r\nYou can divide exponential expressions, leaving the answers as exponential expressions, as long as the bases are the same. To divide exponents (or powers) with the same base, subtract the exponents. Division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.\r\n\r\nFor example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165077.image0.png\" alt=\"image0.png\" width=\"194\" height=\"32\" />\r\n\r\nPretty easy, huh? Now wrap your brain around this:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/165078.image1.png\" alt=\"image1.png\" width=\"406\" height=\"74\" />\r\n<p class=\"Remember\">Any number to the power of zero equals 1, as long as the base number is not 0.</p>","blurb":"","authors":[],"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":"How to multiply exponents","target":"#tab1"},{"label":"How to divide exponents","target":"#tab2"}],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":294686,"title":"Algebra II All-in-One For Dummies Cheat Sheet","slug":"algebra-ii-all-in-one-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/294686"}},{"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"}}]},"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-63a3740e8de96\"></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-63a3740e8e3b7\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-09T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":194356},{"headers":{"creationTime":"2017-05-25T18:45:56+00:00","modifiedTime":"2022-11-04T15:50:20+00:00","timestamp":"2022-11-04T18:01:02+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":"Basic Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"How to Calculate Percentages","strippedTitle":"how to calculate percentages","slug":"how-to-calculate-percentages","canonicalUrl":"","seo":{"metaDescription":"Figure out how much to tip or how good that sale price actually is by learning how to calculate percentages.","noIndex":0,"noFollow":0},"content":"<figure style=\"margin: 0;\"><figcaption style=\"margin-bottom: 10px;\">Listen to the article:</figcaption><audio src=\"/wp-content/uploads/how-to-calculate-percentages.mp3\" controls=\"controls\"><a href=\"/wp-content/uploads/how-to-calculate-percentages.mp3\">Download audio</a></audio></figure>\r\n\r\nWhether you're leaving a tip at a restaurant or figuring out just how much those stylish shoes are <a href=\"https://www.dummies.com/article/academics-the-arts/math/pre-algebra/how-to-calculate-a-percentage-discount-191241/\">on sale</a>, you can't get away from percentages. While there are numerous percentage calculators online, it's helpful to be able to do some quick math in your head to calculate percentages without any digital assistance.\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/percentages.jpg\"><img class=\"aligncenter wp-image-240019 size-full\" src=\"https://www.dummies.com/wp-content/uploads/percentages.jpg\" alt=\"calculating percentages\" width=\"535\" height=\"267\" /></a>\r\n<h2 id=\"tab1\" >What is percentage?</h2>\r\nThe word percentage comes from the word percent. If you split the word percent into its root words, you see “per” and “cent.” Cent is an old European word with French, Latin, and Italian origins meaning “hundred.\" So, percent is translated directly to “per hundred.” If you have 87 percent, you literally have 87 per 100. If it snowed 13 times in the last 100 days, it snowed 13 percent of the time.\r\n<h2 id=\"tab2\" >How to find percentage</h2>\r\nThe numbers that you will be converting into percentages can be given to you in two different formats: decimal and fraction. Decimal format is easier to calculate into a percentage. Converting a decimal to a percentage is as simple as multiplying it by 100. To convert .87 to a percent, simply multiply .87 by 100.\r\n\r\n.87 × 100=87, which gives us 87 percent.\r\n<p class=\"article-tips tip\">Percent is often abbreviated with the % symbol. You can present your answer as 87% or 87 percent — either way is acceptable.</p>\r\nIf you are given a fraction, convert it to a percentage by dividing the top number by the bottom number. If you are given 13/100, you would divide 13 by 100.\r\n\r\n13 ÷ 100 = .13\r\n\r\nThen, follow the steps above for converting a decimal to a percent.\r\n\r\n.13 × 100 = 13, thus giving you 13%.\r\n\r\nThe more difficult task comes when you need to know a percentage when you are given numbers that don’t fit so neatly into 100.\r\n\r\nMost of the time, you will be given a percentage of a specific number. For example, you may know that 40 percent of your paycheck will go to taxes and you want to find out how much money that is.\r\n\r\n \r\n<h2 id=\"tab3\" >How to calculate percentage of a specific number</h2>\r\nThis process is the reverse of what you did earlier. First convert the percentage number to a decimal. Then, you divide your percentage by 100. So, 40 percent would be 40 divided by 100.\r\n\r\n40 ÷ 100 = .40\r\n\r\nNext, once you have the decimal version of your percentage, simply multiply it by the given number (in this case, the amount of your paycheck). If your paycheck is $750, you would multiply 750 by .40.\r\n\r\n750 × .40 = 300\r\n\r\nYour answer would be 300. You are paying $300 in taxes.\r\n\r\nLet’s try another example. You need to save 25 percent of your paycheck for the next 6 months to pay for an upcoming vacation. If your paycheck is $1,500, how much should you save?\r\n\r\nStart by converting 25 percent to a decimal.\r\n\r\n25 ÷ 100 = .25\r\n\r\nNow, multiply the decimal by the amount of your paycheck, or 1500.\r\n\r\n1500 × .25 = 375\r\n\r\nThis means you need to save $375 from each paycheck.","description":"<figure style=\"margin: 0;\"><figcaption style=\"margin-bottom: 10px;\">Listen to the article:</figcaption><audio src=\"/wp-content/uploads/how-to-calculate-percentages.mp3\" controls=\"controls\"><a href=\"/wp-content/uploads/how-to-calculate-percentages.mp3\">Download audio</a></audio></figure>\r\n\r\nWhether you're leaving a tip at a restaurant or figuring out just how much those stylish shoes are <a href=\"https://www.dummies.com/article/academics-the-arts/math/pre-algebra/how-to-calculate-a-percentage-discount-191241/\">on sale</a>, you can't get away from percentages. While there are numerous percentage calculators online, it's helpful to be able to do some quick math in your head to calculate percentages without any digital assistance.\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/percentages.jpg\"><img class=\"aligncenter wp-image-240019 size-full\" src=\"https://www.dummies.com/wp-content/uploads/percentages.jpg\" alt=\"calculating percentages\" width=\"535\" height=\"267\" /></a>\r\n<h2 id=\"tab1\" >What is percentage?</h2>\r\nThe word percentage comes from the word percent. If you split the word percent into its root words, you see “per” and “cent.” Cent is an old European word with French, Latin, and Italian origins meaning “hundred.\" So, percent is translated directly to “per hundred.” If you have 87 percent, you literally have 87 per 100. If it snowed 13 times in the last 100 days, it snowed 13 percent of the time.\r\n<h2 id=\"tab2\" >How to find percentage</h2>\r\nThe numbers that you will be converting into percentages can be given to you in two different formats: decimal and fraction. Decimal format is easier to calculate into a percentage. Converting a decimal to a percentage is as simple as multiplying it by 100. To convert .87 to a percent, simply multiply .87 by 100.\r\n\r\n.87 × 100=87, which gives us 87 percent.\r\n<p class=\"article-tips tip\">Percent is often abbreviated with the % symbol. You can present your answer as 87% or 87 percent — either way is acceptable.</p>\r\nIf you are given a fraction, convert it to a percentage by dividing the top number by the bottom number. If you are given 13/100, you would divide 13 by 100.\r\n\r\n13 ÷ 100 = .13\r\n\r\nThen, follow the steps above for converting a decimal to a percent.\r\n\r\n.13 × 100 = 13, thus giving you 13%.\r\n\r\nThe more difficult task comes when you need to know a percentage when you are given numbers that don’t fit so neatly into 100.\r\n\r\nMost of the time, you will be given a percentage of a specific number. For example, you may know that 40 percent of your paycheck will go to taxes and you want to find out how much money that is.\r\n\r\n \r\n<h2 id=\"tab3\" >How to calculate percentage of a specific number</h2>\r\nThis process is the reverse of what you did earlier. First convert the percentage number to a decimal. Then, you divide your percentage by 100. So, 40 percent would be 40 divided by 100.\r\n\r\n40 ÷ 100 = .40\r\n\r\nNext, once you have the decimal version of your percentage, simply multiply it by the given number (in this case, the amount of your paycheck). If your paycheck is $750, you would multiply 750 by .40.\r\n\r\n750 × .40 = 300\r\n\r\nYour answer would be 300. You are paying $300 in taxes.\r\n\r\nLet’s try another example. You need to save 25 percent of your paycheck for the next 6 months to pay for an upcoming vacation. If your paycheck is $1,500, how much should you save?\r\n\r\nStart by converting 25 percent to a decimal.\r\n\r\n25 ÷ 100 = .25\r\n\r\nNow, multiply the decimal by the amount of your paycheck, or 1500.\r\n\r\n1500 × .25 = 375\r\n\r\nThis means you need to save $375 from each paycheck.","blurb":"","authors":[{"authorId":8941,"name":"Ashley Watters, Abshier House","slug":"ashley-watters-abshier-house","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8941"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"What is percentage?","target":"#tab1"},{"label":"How to find percentage","target":"#tab2"},{"label":"How to calculate percentage of a specific number","target":"#tab3"}],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/253710"}},{"articleId":249996,"title":"Pre-Algebra Practice Questions: Solving Simple Algebraic Equations","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra Practice Questions: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/249986"}},{"articleId":249980,"title":"Pre-Algebra Practice Questions: Rearranging Equations to Isolate x","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/249980"}}]},"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;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6365535e91185\"></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;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6365535e920ec\"></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":"Solve","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-07-07T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":240018},{"headers":{"creationTime":"2016-12-01T01:45:36+00:00","modifiedTime":"2022-10-26T20:26:00+00:00","timestamp":"2022-10-26T21:01:02+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":"Geometry","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33725"},"slug":"geometry","categoryId":33725}],"title":"Calculate the Volume of a Cylinder","strippedTitle":"calculate the volume of a cylinder","slug":"calculate-volume-cylinder","canonicalUrl":"","seo":{"metaDescription":"The volume of an object is how much space the object takes up — or, if you were to drop the object into a full tub of water, how much water would overflow.To ca","noIndex":0,"noFollow":0},"content":"The volume of an object is how much space the object takes up — or, if you were to drop the object into a full tub of water, how much water would overflow.\r\n\r\nTo calculate the volume of a cylinder, you need to know its height and the area of its base. Because a cylinder is a flat-top figure (a solid with two congruent, parallel bases), the base can be either the top or bottom.\r\n<p class=\"article-tips tip\">If you know a cylinder's height and lateral area, but not its radius, you can use the formula for surface area to find the radius, and then calculate the volume from there.</p>\r\n<p class=\"article-tips remember\">The lateral area of a cylinder is basically one rectangle rolled into a tube shape. Think of the lateral area of a cylinder as one rectangular paper towel that rolls exactly once around a paper towel roll. The base of this rectangle (you know, the part of the towel that wraps around the bottom of the roll) is the same as the circumference of the cylinder's base. And the height of the paper towel is the same as the height of the cylinder.</p>\r\n<a href=\"https://www.dummies.com/wp-content/uploads/9781119181552-fg1701cropCyl.gif\"><img class=\"aligncenter wp-image-229752 size-full\" src=\"https://www.dummies.com/wp-content/uploads/9781119181552-fg1701cropCyl.gif\" alt=\"geometry-cylinder\" width=\"329\" height=\"306\" /></a>\r\n<h2 id=\"tab1\" >Use this formula to calculate the volume of a cylinder</h2>\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_01.gif\"><img src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_01.gif\" alt=\"geometry-cylinder-formula\" width=\"184\" height=\"28\" /></a>\r\n\r\nNow for a cylinder problem:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_02.gif\"><img class=\"aligncenter wp-image-229754 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_02.gif\" alt=\"geometry-cylinder-problem\" width=\"436\" height=\"47\" /></a>\r\n\r\nHere's a diagram to help you.\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/9781119181552-un1702.gif\"><img class=\"aligncenter wp-image-229640 size-full\" src=\"https://www.dummies.com/wp-content/uploads/9781119181552-un1702.gif\" alt=\"Geometry-volume-diagram\" width=\"300\" height=\"242\" /></a>\r\n\r\nTo use the volume formula, you need the cylinder's height (which you know) and the area of its base. To get the area of the base, you need its radius. And to get the radius, you can use the surface area formula and solve for <em>r</em>:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_03.gif\"><img class=\"alignnone wp-image-229755 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_03.gif\" alt=\"geometry-radius\" width=\"395\" height=\"61\" /></a>\r\n\r\nRemember that this \"rectangle\" is rolled around the cylinder and that the \"rectangle's\" base is the <a href=\"https://www.dummies.com/article/academics-the-arts/math/basic-math/how-to-measure-circles-149991/\">circumference</a> of the cylinder's circular base. You fill in the equation as follows:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_04.gif\"><img class=\"aligncenter wp-image-229756 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_04.gif\" alt=\"geometry-circumference\" width=\"323\" height=\"119\" /></a>\r\n\r\nNow set the equation equal to zero and factor:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_05.gif\"><img class=\"alignnone wp-image-229757 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_05.gif\" alt=\"geometry-equal-zero\" width=\"173\" height=\"83\" /></a>\r\n\r\nThe radius can't be negative, so it's 5. Now you can finish with the volume formula:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_06.gif\"><img class=\"alignnone wp-image-229758 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_06.gif\" alt=\"geometry-volume-finish\" width=\"184\" height=\"124\" /></a>\r\n\r\nThat does it.","description":"The volume of an object is how much space the object takes up — or, if you were to drop the object into a full tub of water, how much water would overflow.\r\n\r\nTo calculate the volume of a cylinder, you need to know its height and the area of its base. Because a cylinder is a flat-top figure (a solid with two congruent, parallel bases), the base can be either the top or bottom.\r\n<p class=\"article-tips tip\">If you know a cylinder's height and lateral area, but not its radius, you can use the formula for surface area to find the radius, and then calculate the volume from there.</p>\r\n<p class=\"article-tips remember\">The lateral area of a cylinder is basically one rectangle rolled into a tube shape. Think of the lateral area of a cylinder as one rectangular paper towel that rolls exactly once around a paper towel roll. The base of this rectangle (you know, the part of the towel that wraps around the bottom of the roll) is the same as the circumference of the cylinder's base. And the height of the paper towel is the same as the height of the cylinder.</p>\r\n<a href=\"https://www.dummies.com/wp-content/uploads/9781119181552-fg1701cropCyl.gif\"><img class=\"aligncenter wp-image-229752 size-full\" src=\"https://www.dummies.com/wp-content/uploads/9781119181552-fg1701cropCyl.gif\" alt=\"geometry-cylinder\" width=\"329\" height=\"306\" /></a>\r\n<h2 id=\"tab1\" >Use this formula to calculate the volume of a cylinder</h2>\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_01.gif\"><img src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_01.gif\" alt=\"geometry-cylinder-formula\" width=\"184\" height=\"28\" /></a>\r\n\r\nNow for a cylinder problem:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_02.gif\"><img class=\"aligncenter wp-image-229754 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_02.gif\" alt=\"geometry-cylinder-problem\" width=\"436\" height=\"47\" /></a>\r\n\r\nHere's a diagram to help you.\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/9781119181552-un1702.gif\"><img class=\"aligncenter wp-image-229640 size-full\" src=\"https://www.dummies.com/wp-content/uploads/9781119181552-un1702.gif\" alt=\"Geometry-volume-diagram\" width=\"300\" height=\"242\" /></a>\r\n\r\nTo use the volume formula, you need the cylinder's height (which you know) and the area of its base. To get the area of the base, you need its radius. And to get the radius, you can use the surface area formula and solve for <em>r</em>:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_03.gif\"><img class=\"alignnone wp-image-229755 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_03.gif\" alt=\"geometry-radius\" width=\"395\" height=\"61\" /></a>\r\n\r\nRemember that this \"rectangle\" is rolled around the cylinder and that the \"rectangle's\" base is the <a href=\"https://www.dummies.com/article/academics-the-arts/math/basic-math/how-to-measure-circles-149991/\">circumference</a> of the cylinder's circular base. You fill in the equation as follows:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_04.gif\"><img class=\"aligncenter wp-image-229756 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_04.gif\" alt=\"geometry-circumference\" width=\"323\" height=\"119\" /></a>\r\n\r\nNow set the equation equal to zero and factor:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_05.gif\"><img class=\"alignnone wp-image-229757 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_05.gif\" alt=\"geometry-equal-zero\" width=\"173\" height=\"83\" /></a>\r\n\r\nThe radius can't be negative, so it's 5. Now you can finish with the volume formula:\r\n\r\n<a href=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_06.gif\"><img class=\"alignnone wp-image-229758 size-full\" src=\"https://www.dummies.com/wp-content/uploads/GEOM-3E_0017_06.gif\" alt=\"geometry-volume-finish\" width=\"184\" height=\"124\" /></a>\r\n\r\nThat does it.","blurb":"","authors":[{"authorId":8957,"name":"Mark Ryan","slug":"mark-ryan","description":" <p><b>Mark Ryan</b> is the owner of The Math Center in Chicago, Illinois, where he teaches students in all levels of mathematics, from pre&#45;algebra to calculus. He is the author of <i>Calculus For Dummies</i> and <i> Geometry For Dummies.</i> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8957"}}],"primaryCategoryTaxonomy":{"categoryId":33725,"title":"Geometry","slug":"geometry","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33725"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Use this formula to calculate the volume of a cylinder","target":"#tab1"}],"relatedArticles":{"fromBook":[{"articleId":230077,"title":"How to Copy an Angle Using a Compass","slug":"copy-angle-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230077"}},{"articleId":230072,"title":"How to Copy a Line Segment Using a Compass","slug":"copy-line-segment-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230072"}},{"articleId":230069,"title":"How to Find the Right Angle to Two Points","slug":"find-right-angle-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230069"}},{"articleId":230066,"title":"Find the Locus of Points Equidistant from Two Points","slug":"find-locus-points-equidistant-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230066"}},{"articleId":230063,"title":"How to Solve a Two-Dimensional Locus Problem","slug":"solve-two-dimensional-locus-problem","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230063"}}],"fromCategory":[{"articleId":230077,"title":"How to Copy an Angle Using a Compass","slug":"copy-angle-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230077"}},{"articleId":230072,"title":"How to Copy a Line Segment Using a Compass","slug":"copy-line-segment-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230072"}},{"articleId":230069,"title":"How to Find the Right Angle to Two Points","slug":"find-right-angle-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230069"}},{"articleId":230066,"title":"Find the Locus of Points Equidistant from Two Points","slug":"find-locus-points-equidistant-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230066"}},{"articleId":230063,"title":"How to Solve a Two-Dimensional Locus Problem","slug":"solve-two-dimensional-locus-problem","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/230063"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282230,"slug":"geometry-for-dummies-3rd-edition","isbn":"9781119181552","categoryList":["academics-the-arts","math","geometry"],"amazon":{"default":"https://www.amazon.com/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119181550/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/1119181550-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/geometry-for-dummies-3rd-edition-cover-9781119181552-201x255.jpg","width":201,"height":255},"title":"Geometry For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8957\">Mark Ryan </b>is the founder and owner of The Math Center in the Chicago area, where he provides tutoring in all math subjects as well as test preparation. Mark is the author of <i>Calculus For Dummies, Calculus Workbook For Dummies</i>, and <i>Geometry Workbook For Dummies</i>.</p>","authors":[{"authorId":8957,"name":"Mark Ryan","slug":"mark-ryan","description":" <p><b>Mark Ryan</b> is the owner of The Math Center in Chicago, Illinois, where he teaches students in all levels of mathematics, from pre&#45;algebra to calculus. He is the author of <i>Calculus For Dummies</i> and <i> Geometry For Dummies.</i> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8957"}}],"_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;geometry&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119181552&quot;]}]\" id=\"du-slot-6359a00e7da10\"></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;geometry&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119181552&quot;]}]\" id=\"du-slot-6359a00e7e36a\"></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":"Explore","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-10-26T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":229750},{"headers":{"creationTime":"2016-03-26T08:26:22+00:00","modifiedTime":"2022-10-26T15:27:35+00:00","timestamp":"2022-10-26T18:01:02+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":"Statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"},"slug":"statistics","categoryId":33728}],"title":"How to Use the T-table to Solve Statistics Problems","strippedTitle":"how to use the t-table to solve statistics problems","slug":"how-to-use-the-t-table-to-solve-statistics-problems","canonicalUrl":"","seo":{"metaDescription":"How exactly does a t-table differ from a z-table? Learn about all the important statistical differences here.","noIndex":0,"noFollow":0},"content":"The t<i>-</i>table (for the t<i>-</i>distribution) is different from the <i>z-</i>table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various <a href=\"http://dummies.com/article/academics-the-arts/math/statistics/statistical-t-distribution-the-t-table-190873/\">t-distributions</a>, using the t<i>-</i>table, is a valuable statistics skill.\r\n<h2 id=\"tab1\" >How to use the t-table to find right-tail probabilities and p-values for hypothesis tests involving t:</h2>\r\n<ol>\r\n \t<li>First, find the t-value for which you want the right-tail probability (call it t), and find the sample size (for example, n).</li>\r\n \t<li>Next, find the row corresponding to the degrees of freedom (df) for your problem (for example, n – 1). Go across that row to find the two t-values between which your t falls. For example, if your t is 1.60 and your n is 7, you look in the row for df = 7 – 1 = 6. Across that row you find your t lies between t-values 1.44 and 1.94.</li>\r\n \t<li>Then, go to the top of the columns containing the two t-values from Step 2. The right-tail (greater-than) probability for your t-value is somewhere between the two values at the top of these columns. For example, your t = 1.60 is between t-values 1.44 and 1.94 (df = 6); so the right tail probability for your t is between 0.10 (column heading for t = 1.44); and 0.05 (column heading for t = 1.94).</li>\r\n</ol>\r\n \r\n<p class=\"article-tips remember\">\r\nThe row near the bottom with Z in the df column gives right-tail (greater-than) probabilities from the Z-distribution.</p>\r\n\r\nUse the t table to find t*-values (critical values) for a confidence interval involving t:\r\n<ol>\r\n \t<li>Determine the confidence level you need (as a percentage).</li>\r\n \t<li>Determine the sample size (for example, n).</li>\r\n \t<li>Look at the bottom row of the table where the percentages are shown. Find your % confidence level there.</li>\r\n \t<li>Intersect this column with the row representing your degrees of freedom (df).</li>\r\n</ol>\r\nThis is the t-value you need for your confidence interval. For example, a 95% confidence interval with df=6 has t*=2.45. (Find 95% on the last line and go up to row 6.)\r\n\r\n \r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451675.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"796\" />\r\n\r\n \r\n<h2 id=\"tab2\" >Practice solving problems using the t-table sample questions below</h2>\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 17</p>\r\n<p class=\"child-para\">The study involving one population and a sample size of 18 has <i>n</i> – 1 = 18 – 1 = 17 degrees of freedom.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">For a study involving a paired design with a total of 44 observations, with the results assuming a t<i>-</i>distribution, what row of the table will you use to find the probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 21</p>\r\n<p class=\"child-para\">A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: <i>n</i> – 1 = 22 – 1 = 21.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">A <i>t-</i>value of 2.35, from a <i>t-</i>distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table?</p>\r\n<p class=\"child-para\"><b>Answer: </b>0.025 and 0.01</p>\r\n<p class=\"child-para\">Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01.</p>\r\n<p class=\"child-para\">Hence, the upper-tail probability for a <i>t-</i>value of 2.35 must lie between 0.025 and 0.01.</p>\r\n</li>\r\n</ol>","description":"The t<i>-</i>table (for the t<i>-</i>distribution) is different from the <i>z-</i>table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various <a href=\"http://dummies.com/article/academics-the-arts/math/statistics/statistical-t-distribution-the-t-table-190873/\">t-distributions</a>, using the t<i>-</i>table, is a valuable statistics skill.\r\n<h2 id=\"tab1\" >How to use the t-table to find right-tail probabilities and p-values for hypothesis tests involving t:</h2>\r\n<ol>\r\n \t<li>First, find the t-value for which you want the right-tail probability (call it t), and find the sample size (for example, n).</li>\r\n \t<li>Next, find the row corresponding to the degrees of freedom (df) for your problem (for example, n – 1). Go across that row to find the two t-values between which your t falls. For example, if your t is 1.60 and your n is 7, you look in the row for df = 7 – 1 = 6. Across that row you find your t lies between t-values 1.44 and 1.94.</li>\r\n \t<li>Then, go to the top of the columns containing the two t-values from Step 2. The right-tail (greater-than) probability for your t-value is somewhere between the two values at the top of these columns. For example, your t = 1.60 is between t-values 1.44 and 1.94 (df = 6); so the right tail probability for your t is between 0.10 (column heading for t = 1.44); and 0.05 (column heading for t = 1.94).</li>\r\n</ol>\r\n \r\n<p class=\"article-tips remember\">\r\nThe row near the bottom with Z in the df column gives right-tail (greater-than) probabilities from the Z-distribution.</p>\r\n\r\nUse the t table to find t*-values (critical values) for a confidence interval involving t:\r\n<ol>\r\n \t<li>Determine the confidence level you need (as a percentage).</li>\r\n \t<li>Determine the sample size (for example, n).</li>\r\n \t<li>Look at the bottom row of the table where the percentages are shown. Find your % confidence level there.</li>\r\n \t<li>Intersect this column with the row representing your degrees of freedom (df).</li>\r\n</ol>\r\nThis is the t-value you need for your confidence interval. For example, a 95% confidence interval with df=6 has t*=2.45. (Find 95% on the last line and go up to row 6.)\r\n\r\n \r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/451675.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"796\" />\r\n\r\n \r\n<h2 id=\"tab2\" >Practice solving problems using the t-table sample questions below</h2>\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 17</p>\r\n<p class=\"child-para\">The study involving one population and a sample size of 18 has <i>n</i> – 1 = 18 – 1 = 17 degrees of freedom.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">For a study involving a paired design with a total of 44 observations, with the results assuming a t<i>-</i>distribution, what row of the table will you use to find the probability affiliated with the study results?</p>\r\n<p class=\"child-para\"><b>Answer:</b> <i>df</i> = 21</p>\r\n<p class=\"child-para\">A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: <i>n</i> – 1 = 22 – 1 = 21.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">A <i>t-</i>value of 2.35, from a <i>t-</i>distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table?</p>\r\n<p class=\"child-para\"><b>Answer: </b>0.025 and 0.01</p>\r\n<p class=\"child-para\">Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01.</p>\r\n<p class=\"child-para\">Hence, the upper-tail probability for a <i>t-</i>value of 2.35 must lie between 0.025 and 0.01.</p>\r\n</li>\r\n</ol>","blurb":"","authors":[{"authorId":8947,"name":"The Experts at Dummies","slug":"the-experts-at-dummies","description":"The Experts at Dummies are smart, friendly people who make learning easy by taking a not-so-serious approach to serious stuff.","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8947"}}],"primaryCategoryTaxonomy":{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33728"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"How to use the t-table to find right-tail probabilities and p-values for hypothesis tests involving t:","target":"#tab1"},{"label":"Practice solving problems using the t-table sample questions below","target":"#tab2"}],"relatedArticles":{"fromBook":[{"articleId":207668,"title":"Statistics: 1001 Practice Problems For Dummies Cheat 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Problems","slug":"sticking-to-a-strategy-when-you-solve-statistics-problems","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/151934"}}],"fromCategory":[{"articleId":263501,"title":"10 Steps to a Better Math Grade with Statistics","slug":"10-steps-to-a-better-math-grade-with-statistics","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263501"}},{"articleId":263495,"title":"Statistics and Histograms","slug":"statistics-and-histograms","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263495"}},{"articleId":263492,"title":"What is Categorical Data and How is It Summarized?","slug":"what-is-categorical-data-and-how-is-it-summarized","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/263492"}},{"articleId":209320,"title":"Statistics II For Dummies Cheat Sheet","slug":"statistics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","statistics"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/209320"}},{"articleId":209293,"title":"SPSS For Dummies Cheat 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1001 Practice Problems For Dummies (+ Free Online Practice)","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"","authors":[{"authorId":34784,"name":"","slug":"","description":" <p>The <b>American Diabetes Association</b> leads the fight against the deadly consequences of diabetes by funding research, delivering services to communities affected by diabetes, and providing objective and credible information. It is led by a network of more than one million volunteers and nearly 14,000 healthcare professionals. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/34784"}}],"_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;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883593&quot;]}]\" id=\"du-slot-635975deca31f\"></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;statistics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119883593&quot;]}]\" id=\"du-slot-635975deca821\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2021-09-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":147282},{"headers":{"creationTime":"2016-03-26T11:06:41+00:00","modifiedTime":"2022-10-24T14:16:10+00:00","timestamp":"2022-10-24T15:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33726"},"slug":"pre-algebra","categoryId":33726}],"title":"10 Alternative Numeral and Number Systems","strippedTitle":"10 alternative numeral and number systems","slug":"10-alternative-numeral-and-number-systems","canonicalUrl":"","seo":{"metaDescription":"The distinction between numbers and numerals is subtle but important. A number is an idea that expresses how much or how many. A numeral is a written symbol tha","noIndex":0,"noFollow":0},"content":"<p>The distinction between numbers and numerals is subtle but important. A number is an idea that expresses how much or how many. A numeral is a written symbol that expresses a number. Here are ten ways to represent numbers that differ from the Hindu-Arabic (decimal) system.</p>\r\n<h2 id=\"tab1\" >Tally marks</h2>\r\n<p>Numbers are abstractions that stand for real things. The first known numbers came into being with the rise of trading and commerce — people needed to keep track of commodities such as animals, harvested crops, or tools. At first, traders used clay or stone tokens to help simplify the job of counting. Over time, tally marks scratched either in bone or on clay took the place of tokens.</p>\r\n<h2 id=\"tab2\" >Bundled tally marks</h2>\r\n<p>As early humans grew more comfortable letting tally marks stand for real-world objects, the next development in numbers was probably tally marks scratched in <i>bundles</i> of 5 (fingers on one hand), 10 (fingers on both hands), or 20 (fingers and toes). Bundling provided a simple way to count larger numbers more easily.</p>\r\n<p>Of course, this system is much easier to read than non-bundled scratches — you can easily multiply or count by fives to get the total. Even today, people keep track of points in games using bundles such as these.</p>\r\n<h2 id=\"tab3\" >Egyptian numerals</h2>\r\n<p>Ancient Egyptian numerals are among the oldest number systems still in use today. Egyptian numerals use seven symbols.</p>\r\n<h3>Egyptian Numerals</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Number</th>\r\n<th>Symbol</th>\r\n</tr>\r\n<tr>\r\n<td>1</td>\r\n<td>Stroke</td>\r\n</tr>\r\n<tr>\r\n<td>10</td>\r\n<td>Yoke</td>\r\n</tr>\r\n<tr>\r\n<td>100</td>\r\n<td>Coil of rope</td>\r\n</tr>\r\n<tr>\r\n<td>1,000</td>\r\n<td>Lotus</td>\r\n</tr>\r\n<tr>\r\n<td>10,000</td>\r\n<td>Finger</td>\r\n</tr>\r\n<tr>\r\n<td>100,000</td>\r\n<td>Frog</td>\r\n</tr>\r\n<tr>\r\n<td>1,000,000</td>\r\n<td>Man with raised hands</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>Numbers are formed by accumulating enough of the symbols that you need. For example,</p>\r\n<p>7 = 7 strokes</p>\r\n<p>24 = 2 yokes, 4 strokes</p>\r\n<p>1,536 = 1 lotus, 5 coils of rope, 3 yokes, 6 strokes</p>\r\n<h2 id=\"tab4\" >Babylonian numerals</h2>\r\n<p>Babylonian numerals, which came into being about 4,000 years ago, use two symbols:</p>\r\n<p>1 = Y</p>\r\n<p>10 = <</p>\r\n<p>For numbers less than 60, numbers are formed by accumulating enough of the symbols you need. For example,</p>\r\n<p>6 = YYYYYY</p>\r\n<p class=\"code\">34 = <<<YYYY</p>\r\n<p>For numbers 60 and beyond, Babylonian numerals use place value based on the number 60.</p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td class=\"code\">61 = Y Y</td>\r\n<td>(one 60 and one 1)</td>\r\n</tr>\r\n<tr>\r\n<td class=\"code\">124 = YY YYYY</td>\r\n<td>(two 60s and four 1s)</td>\r\n</tr>\r\n<tr>\r\n<td class=\"code\">611 = < <Y</td>\r\n<td>(ten 60s and eleven 1s)</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<h2 id=\"tab5\" >Ancient Greek numerals</h2>\r\n<p>Ancient Greek numerals were based on the Greek letters. The numbers from 1 to 999 were formed using the symbols shown:</p>\r\n<h2 id=\"tab6\" >Roman numerals</h2>\r\n<p>Although Roman numerals are over 2,000 years old, people still use them today, either decoratively (for example, on clocks, cornerstones, and Super Bowl memorabilia) or when numerals distinct from decimal numbers are needed (for example, in outlines). Roman numerals use seven symbols, all of which are capital letters in the Latin alphabet (which pretty much happens to be the English alphabet as well):</p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>I = 1</td>\r\n<td>V = 5</td>\r\n<td>X = 10</td>\r\n<td>L = 50</td>\r\n</tr>\r\n<tr>\r\n<td>C = 100</td>\r\n<td>D = 500</td>\r\n<td>M = 1,000</td>\r\n<td></td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<h2 id=\"tab7\" >Mayan numerals</h2>\r\n<p>Mayan numerals developed in South America during roughly the same period that Roman numerals developed in Europe. Mayan numerals use two symbols: dots and horizontal bars. A bar is equal to 5, and a dot is equal to 1. Numbers from 1 to 19 are formed by accumulating dots and bars. For example,</p>\r\n<p>3 = 3 dots</p>\r\n<p>7 = 2 dots over 1 bar</p>\r\n<p>19 = 4 dots over 3 bars</p>\r\n<p>Numbers from 20 to 399 are formed using these same combinations, but raised up to indicate place value. For example,</p>\r\n<p>21 = raised 1 dot, 1 dot (one 20 + one 1)</p>\r\n<p>399 = raised 4 dots over 3 bars, 4 dots over 3 bars (nineteen 20s + three 5s + four 1s)</p>\r\n<h2 id=\"tab8\" >Base-2 (binary) numbers</h2>\r\n<p>Binary numbers use only two symbols: 0 and 1. This simplicity makes binary numbers useful as the number system that computers use for data storage and computation.</p>\r\n<p>Like the decimal system you're most familiar with, binary numbers use place value. Unlike the decimal system, binary place value is based not on powers of ten (1, 10, 100, 1,000, and so forth) but on powers of two (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and so on), as seen here:</p>\r\n<h3>Binary Place Values</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>512s</td>\r\n<td>256s</td>\r\n<td>128s</td>\r\n<td>64s</td>\r\n<td>32s</td>\r\n<td>16s</td>\r\n<td>8s</td>\r\n<td>4s</td>\r\n<td>2s</td>\r\n<td>1s</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<h2 id=\"tab9\" >Base-16 (hexadecimal) numbers</h2>\r\n<p>The computer's first language is binary numbers. But in practice, humans find binary numbers of any significant length virtually undecipherable. Hexadecimal numbers, however, are readable to humans and still easily translated into binary numbers, so computer programmers use hexadecimal numbers as a sort of common language when interfacing with computers at the deepest level, the level of hardware and software design.</p>\r\n<p>The hexadecimal number system uses all ten digits 0 through 9 from the decimal system. Additionally, it uses six more symbols:</p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A = 10</td>\r\n<td>B = 11</td>\r\n<td>C = 12</td>\r\n</tr>\r\n<tr>\r\n<td>D = 13</td>\r\n<td>E = 14</td>\r\n<td>F = 15</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>Hexadecimal is a place-value system based on powers of 16.</p>\r\n<h3>Hexadecimal Place Values</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>1,048,576s</td>\r\n<td>65,536s</td>\r\n<td>4,096s</td>\r\n<td>256s</td>\r\n<td>16s</td>\r\n<td>1s</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>As you can see, each number in the table is exactly 16 times the number to its immediate right.</p>\r\n<h2 id=\"tab10\" >Prime-based numbers</h2>\r\n<p>One wacky way to represent numbers unlike any of the others is prime-based numbers. Prime-based numbers are similar to decimal, binary, and hexadecimal numbers in that they use place value to determine the value of digits. But unlike these other number systems, prime-based numbers are based not on addition but on multiplication.</p>\r\n<h3>Prime-Based Place Values</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>31s</td>\r\n<td>29s</td>\r\n<td>23s</td>\r\n<td>19s</td>\r\n<td>17s</td>\r\n<td>13s</td>\r\n<td>11s</td>\r\n<td>7s</td>\r\n<td>5s</td>\r\n<td>3s</td>\r\n<td>2s</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>You can use the table to find the decimal value of a prime-based number.</p>","description":"<p>The distinction between numbers and numerals is subtle but important. A number is an idea that expresses how much or how many. A numeral is a written symbol that expresses a number. Here are ten ways to represent numbers that differ from the Hindu-Arabic (decimal) system.</p>\r\n<h2 id=\"tab1\" >Tally marks</h2>\r\n<p>Numbers are abstractions that stand for real things. The first known numbers came into being with the rise of trading and commerce — people needed to keep track of commodities such as animals, harvested crops, or tools. At first, traders used clay or stone tokens to help simplify the job of counting. Over time, tally marks scratched either in bone or on clay took the place of tokens.</p>\r\n<h2 id=\"tab2\" >Bundled tally marks</h2>\r\n<p>As early humans grew more comfortable letting tally marks stand for real-world objects, the next development in numbers was probably tally marks scratched in <i>bundles</i> of 5 (fingers on one hand), 10 (fingers on both hands), or 20 (fingers and toes). Bundling provided a simple way to count larger numbers more easily.</p>\r\n<p>Of course, this system is much easier to read than non-bundled scratches — you can easily multiply or count by fives to get the total. Even today, people keep track of points in games using bundles such as these.</p>\r\n<h2 id=\"tab3\" >Egyptian numerals</h2>\r\n<p>Ancient Egyptian numerals are among the oldest number systems still in use today. Egyptian numerals use seven symbols.</p>\r\n<h3>Egyptian Numerals</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Number</th>\r\n<th>Symbol</th>\r\n</tr>\r\n<tr>\r\n<td>1</td>\r\n<td>Stroke</td>\r\n</tr>\r\n<tr>\r\n<td>10</td>\r\n<td>Yoke</td>\r\n</tr>\r\n<tr>\r\n<td>100</td>\r\n<td>Coil of rope</td>\r\n</tr>\r\n<tr>\r\n<td>1,000</td>\r\n<td>Lotus</td>\r\n</tr>\r\n<tr>\r\n<td>10,000</td>\r\n<td>Finger</td>\r\n</tr>\r\n<tr>\r\n<td>100,000</td>\r\n<td>Frog</td>\r\n</tr>\r\n<tr>\r\n<td>1,000,000</td>\r\n<td>Man with raised hands</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>Numbers are formed by accumulating enough of the symbols that you need. For example,</p>\r\n<p>7 = 7 strokes</p>\r\n<p>24 = 2 yokes, 4 strokes</p>\r\n<p>1,536 = 1 lotus, 5 coils of rope, 3 yokes, 6 strokes</p>\r\n<h2 id=\"tab4\" >Babylonian numerals</h2>\r\n<p>Babylonian numerals, which came into being about 4,000 years ago, use two symbols:</p>\r\n<p>1 = Y</p>\r\n<p>10 = <</p>\r\n<p>For numbers less than 60, numbers are formed by accumulating enough of the symbols you need. For example,</p>\r\n<p>6 = YYYYYY</p>\r\n<p class=\"code\">34 = <<<YYYY</p>\r\n<p>For numbers 60 and beyond, Babylonian numerals use place value based on the number 60.</p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td class=\"code\">61 = Y Y</td>\r\n<td>(one 60 and one 1)</td>\r\n</tr>\r\n<tr>\r\n<td class=\"code\">124 = YY YYYY</td>\r\n<td>(two 60s and four 1s)</td>\r\n</tr>\r\n<tr>\r\n<td class=\"code\">611 = < <Y</td>\r\n<td>(ten 60s and eleven 1s)</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<h2 id=\"tab5\" >Ancient Greek numerals</h2>\r\n<p>Ancient Greek numerals were based on the Greek letters. The numbers from 1 to 999 were formed using the symbols shown:</p>\r\n<h2 id=\"tab6\" >Roman numerals</h2>\r\n<p>Although Roman numerals are over 2,000 years old, people still use them today, either decoratively (for example, on clocks, cornerstones, and Super Bowl memorabilia) or when numerals distinct from decimal numbers are needed (for example, in outlines). Roman numerals use seven symbols, all of which are capital letters in the Latin alphabet (which pretty much happens to be the English alphabet as well):</p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>I = 1</td>\r\n<td>V = 5</td>\r\n<td>X = 10</td>\r\n<td>L = 50</td>\r\n</tr>\r\n<tr>\r\n<td>C = 100</td>\r\n<td>D = 500</td>\r\n<td>M = 1,000</td>\r\n<td></td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<h2 id=\"tab7\" >Mayan numerals</h2>\r\n<p>Mayan numerals developed in South America during roughly the same period that Roman numerals developed in Europe. Mayan numerals use two symbols: dots and horizontal bars. A bar is equal to 5, and a dot is equal to 1. Numbers from 1 to 19 are formed by accumulating dots and bars. For example,</p>\r\n<p>3 = 3 dots</p>\r\n<p>7 = 2 dots over 1 bar</p>\r\n<p>19 = 4 dots over 3 bars</p>\r\n<p>Numbers from 20 to 399 are formed using these same combinations, but raised up to indicate place value. For example,</p>\r\n<p>21 = raised 1 dot, 1 dot (one 20 + one 1)</p>\r\n<p>399 = raised 4 dots over 3 bars, 4 dots over 3 bars (nineteen 20s + three 5s + four 1s)</p>\r\n<h2 id=\"tab8\" >Base-2 (binary) numbers</h2>\r\n<p>Binary numbers use only two symbols: 0 and 1. This simplicity makes binary numbers useful as the number system that computers use for data storage and computation.</p>\r\n<p>Like the decimal system you're most familiar with, binary numbers use place value. Unlike the decimal system, binary place value is based not on powers of ten (1, 10, 100, 1,000, and so forth) but on powers of two (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and so on), as seen here:</p>\r\n<h3>Binary Place Values</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>512s</td>\r\n<td>256s</td>\r\n<td>128s</td>\r\n<td>64s</td>\r\n<td>32s</td>\r\n<td>16s</td>\r\n<td>8s</td>\r\n<td>4s</td>\r\n<td>2s</td>\r\n<td>1s</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<h2 id=\"tab9\" >Base-16 (hexadecimal) numbers</h2>\r\n<p>The computer's first language is binary numbers. But in practice, humans find binary numbers of any significant length virtually undecipherable. Hexadecimal numbers, however, are readable to humans and still easily translated into binary numbers, so computer programmers use hexadecimal numbers as a sort of common language when interfacing with computers at the deepest level, the level of hardware and software design.</p>\r\n<p>The hexadecimal number system uses all ten digits 0 through 9 from the decimal system. Additionally, it uses six more symbols:</p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>A = 10</td>\r\n<td>B = 11</td>\r\n<td>C = 12</td>\r\n</tr>\r\n<tr>\r\n<td>D = 13</td>\r\n<td>E = 14</td>\r\n<td>F = 15</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>Hexadecimal is a place-value system based on powers of 16.</p>\r\n<h3>Hexadecimal Place Values</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>1,048,576s</td>\r\n<td>65,536s</td>\r\n<td>4,096s</td>\r\n<td>256s</td>\r\n<td>16s</td>\r\n<td>1s</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>As you can see, each number in the table is exactly 16 times the number to its immediate right.</p>\r\n<h2 id=\"tab10\" >Prime-based numbers</h2>\r\n<p>One wacky way to represent numbers unlike any of the others is prime-based numbers. Prime-based numbers are similar to decimal, binary, and hexadecimal numbers in that they use place value to determine the value of digits. But unlike these other number systems, prime-based numbers are based not on addition but on multiplication.</p>\r\n<h3>Prime-Based Place Values</h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>31s</td>\r\n<td>29s</td>\r\n<td>23s</td>\r\n<td>19s</td>\r\n<td>17s</td>\r\n<td>13s</td>\r\n<td>11s</td>\r\n<td>7s</td>\r\n<td>5s</td>\r\n<td>3s</td>\r\n<td>2s</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p>You can use the table to find the decimal value of a prime-based number.</p>","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is an instructor and math and test prep tutor in New Jersey. He is the author of <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, ACT Math For Dummies, Logic For Dummies,</i> and <i>Calculus II For Dummies</i>. In his spare time, he enjoys traveling and learning foreign languages.</p> ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33726"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Tally marks","target":"#tab1"},{"label":"Bundled tally marks","target":"#tab2"},{"label":"Egyptian numerals","target":"#tab3"},{"label":"Babylonian numerals","target":"#tab4"},{"label":"Ancient Greek numerals","target":"#tab5"},{"label":"Roman numerals","target":"#tab6"},{"label":"Mayan numerals","target":"#tab7"},{"label":"Base-2 (binary) numbers","target":"#tab8"},{"label":"Base-16 (hexadecimal) numbers","target":"#tab9"},{"label":"Prime-based 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\"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6356a8aecd821\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;pre-algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" 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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. ","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":289189,"slug":"algebra-i-all-in-one-for-dummies","isbn":"9781119843047","categoryList":["academics-the-arts","math","algebra"],"amazon":{"default":"https://www.amazon.com/gp/product/1119843049/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119843049/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/1119843049-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119843049/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119843049/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/1119843049-204x255.jpg","width":204,"height":255},"title":"Algebra I All-in-One For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"8985\">Mary Jane Sterling</b></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.</p>","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. ","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":"2016-03-26T15:11:52+00:00","modifiedTime":"2022-10-06T21:01:24+00:00","timestamp":"2022-10-07T00:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Pre-Calculus: Test the Roots By Long Division of Polynomials","strippedTitle":"pre-calculus: test the roots by long division of polynomials","slug":"how-to-guess-and-check-real-roots-2-testing-roots-by-dividing-polynomials-using-long-division","canonicalUrl":"","seo":{"metaDescription":"One way to test the real roots is to use long division of polynomials and hope that when you divide you get a remainder of 0.","noIndex":0,"noFollow":0},"content":"Once you have used the rational root theorem to list all the possible<i> </i>rational roots of any polynomial, the next step is to test the roots. One way is to use long division of polynomials and hope that when you divide you get a remainder of 0. Once you have a list of possible rational roots, you then pick one and assume that it’s a root.\r\n\r\nFor example, consider the equation <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48, which has the following possible rational roots:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370082.image0.png\" alt=\"image0.png\" width=\"453\" height=\"19\" />\r\n\r\nIf <i>x</i> = <i>c</i> is a root<i>,</i> then <i>x</i> – c is a factor. So if you pick <i>x</i> = 2 as your guess for the root, <i>x</i> – 2 should be a factor. You can use long division to test if <i>x</i> – 2 is actually a factor and, therefore, <i>x</i> = 2 is a root.\r\n\r\nDividing polynomials to get a specific answer isn’t something you do every day, but the idea of a function or expression that’s written as the quotient of two polynomials is important for pre-calculus. If you divide a polynomial by another and get a remainder of 0, the divisor is a factor, which in turn gives a root.\r\n<p class=\"TechnicalStuff\">In math lingo, the division algorithm states the following: If <i>f</i>(<i>x</i>) and <i>d</i>(<i>x</i>) are polynomials such that <i>d</i>(<i>x</i>) isn’t equal to 0, and the degree of <i>d</i>(<i>x</i>) isn’t larger than the degree of <i>f</i>(<i>x</i>), there are unique polynomials <i>q</i>(<i>x</i>) and <i>r</i>(<i>x</i>)<i> </i>such that</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370083.image1.png\" alt=\"image1.png\" width=\"159\" height=\"20\" />\r\n\r\nIn plain English, the dividend equals the divisor times the quotient plus the remainder. You can always check your results by remembering this information.\r\n<p class=\"Tip\">Remember the mnemonic device<i> </i><u>D</u>irty <u>M</u>onkeys <u>S</u>mell <u>B</u>ad when doing long division to check your roots. Make sure all terms in the polynomial are listed in descending order and that every degree is represented. In other words, if <i>x</i><sup>2</sup> is missing, put in a placeholder of 0<i>x</i><sup>2</sup> and then do the division. (This step is just to make the division process easier.)</p>\r\nTo divide two polynomials, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">Divide the leading term of the dividend by the leading term of the divisor. Write this quotient directly above the term you just divided into.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply the quotient term from Step 1 by the entire divisor. Write this polynomial under the dividend so that like terms are lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract the whole line you just wrote from the dividend.</p>\r\n<p class=\"child-para\">You can change all the signs and add if it makes you feel more comfortable. This way, you won’t forget signs.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down the next term.</p>\r\n<p class=\"child-para\">Do exactly what this says; bring down the next term in the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Repeat Steps 1–4 over and over until the remainder polynomial has a degree that’s less than the dividend’s.</p>\r\n</li>\r\n</ol>\r\n<p class=\"Tip\">The following list explains how to divide 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x </i>+ 48 by <i>x</i> – 2. Each step corresponds with the numbered step in the illustration in this figure.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370084.image2.jpg\" alt=\"The process of long division of polynomials.\" width=\"400\" height=\"394\" />\r\n<div class=\"imageCaption\">The process of long division of polynomials.</div>\r\n</div>\r\n(Note that using Descartes’s rule of signs, you find that this particular example may have positive roots, so it’s efficient to try a positive number here. If Descartes’s rule of signs had said that no positive roots existed, you wouldn’t test any positives!)\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">What do you have to multiply <i>x</i> in the divisor by to make it become 2<i>x</i><sup>4</sup> in the dividend? The quotient, 2<i>x</i><sup>3</sup>, goes above the 2<i>x</i><sup>4</sup> term.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply this quotient by the divisor and write it under the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract this line from the dividend: (2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup>) – (2<i>x</i><sup>4</sup> – 4<i>x</i><sup>3</sup>) = –5<i>x</i><sup>3</sup>. If you’ve done the job right, the subtraction of the first terms always produces 0.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\">Bring down the other terms of the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What do you have to multiply <i>x</i> by to make it –5<i>x</i><sup>3</sup>? Put the answer, –5<i>x</i><sup>2</sup>, above the –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>Multiply the –5<i>x</i><sup>2</sup> times the <i>x</i> – 2 to get –5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>. Write it under the remainder with the degrees lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You now have (–5<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup>) – (–5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>) = –31<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +88<i>x</i> takes its place.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What to multiply by to make <i>x</i> become –31<i>x</i><sup>2</sup>? The quotient –31<i>x</i> goes above –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The value –31<i>x</i> times (<i>x</i> – 2) is –31<i>x</i><sup>2</sup> + 62<i>x;</i> write it under the remainder.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b> You now have (–31<i>x</i><sup>2</sup> + 88<i>x</i>) – (–31<i>x</i><sup>2</sup> + 62<i>x</i>), which is 26<i>x.</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +48 comes down.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>The term 26<i>x</i> divided by <i>x</i> is 26. This answer goes on top.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The constant 26 multiplied by (<i>x</i> – 2) is 26<i>x</i> – 52.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You subtract (26<i>x</i> + 48) – (26<i>x</i> – 52) to get 100.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>top.</p>\r\n<p class=\"child-para\"><b></b>The remainder 100 has a degree that’s less than the divisor of <i>x</i> – 2.</p>\r\n</li>\r\n</ol>\r\nWow . . . now you know why they call it <i>long</i> division. You went through all that to find out that <i>x</i> – 2 isn’t a factor of the polynomial, which means that <i>x</i> = 2 isn’t a root.\r\n<p class=\"Remember\">If you divide by <i>c</i> and the remainder is 0, then the linear expression (<i>x</i> – <i>c</i>) is a factor and that <i>c</i> is a root. A remainder other than 0 implies that (<i>x</i> – <i>c</i>) isn’t a factor and that <i>c</i> isn’t a root.</p>","description":"Once you have used the rational root theorem to list all the possible<i> </i>rational roots of any polynomial, the next step is to test the roots. One way is to use long division of polynomials and hope that when you divide you get a remainder of 0. Once you have a list of possible rational roots, you then pick one and assume that it’s a root.\r\n\r\nFor example, consider the equation <i>f</i>(<i>x</i>) = 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x</i> + 48, which has the following possible rational roots:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370082.image0.png\" alt=\"image0.png\" width=\"453\" height=\"19\" />\r\n\r\nIf <i>x</i> = <i>c</i> is a root<i>,</i> then <i>x</i> – c is a factor. So if you pick <i>x</i> = 2 as your guess for the root, <i>x</i> – 2 should be a factor. You can use long division to test if <i>x</i> – 2 is actually a factor and, therefore, <i>x</i> = 2 is a root.\r\n\r\nDividing polynomials to get a specific answer isn’t something you do every day, but the idea of a function or expression that’s written as the quotient of two polynomials is important for pre-calculus. If you divide a polynomial by another and get a remainder of 0, the divisor is a factor, which in turn gives a root.\r\n<p class=\"TechnicalStuff\">In math lingo, the division algorithm states the following: If <i>f</i>(<i>x</i>) and <i>d</i>(<i>x</i>) are polynomials such that <i>d</i>(<i>x</i>) isn’t equal to 0, and the degree of <i>d</i>(<i>x</i>) isn’t larger than the degree of <i>f</i>(<i>x</i>), there are unique polynomials <i>q</i>(<i>x</i>) and <i>r</i>(<i>x</i>)<i> </i>such that</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370083.image1.png\" alt=\"image1.png\" width=\"159\" height=\"20\" />\r\n\r\nIn plain English, the dividend equals the divisor times the quotient plus the remainder. You can always check your results by remembering this information.\r\n<p class=\"Tip\">Remember the mnemonic device<i> </i><u>D</u>irty <u>M</u>onkeys <u>S</u>mell <u>B</u>ad when doing long division to check your roots. Make sure all terms in the polynomial are listed in descending order and that every degree is represented. In other words, if <i>x</i><sup>2</sup> is missing, put in a placeholder of 0<i>x</i><sup>2</sup> and then do the division. (This step is just to make the division process easier.)</p>\r\nTo divide two polynomials, follow these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">Divide the leading term of the dividend by the leading term of the divisor. Write this quotient directly above the term you just divided into.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply the quotient term from Step 1 by the entire divisor. Write this polynomial under the dividend so that like terms are lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract the whole line you just wrote from the dividend.</p>\r\n<p class=\"child-para\">You can change all the signs and add if it makes you feel more comfortable. This way, you won’t forget signs.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down the next term.</p>\r\n<p class=\"child-para\">Do exactly what this says; bring down the next term in the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Repeat Steps 1–4 over and over until the remainder polynomial has a degree that’s less than the dividend’s.</p>\r\n</li>\r\n</ol>\r\n<p class=\"Tip\">The following list explains how to divide 2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup> + 88<i>x </i>+ 48 by <i>x</i> – 2. Each step corresponds with the numbered step in the illustration in this figure.</p>\r\n\r\n<div class=\"imageBlock\" style=\"width: 400px;\">\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/370084.image2.jpg\" alt=\"The process of long division of polynomials.\" width=\"400\" height=\"394\" />\r\n<div class=\"imageCaption\">The process of long division of polynomials.</div>\r\n</div>\r\n(Note that using Descartes’s rule of signs, you find that this particular example may have positive roots, so it’s efficient to try a positive number here. If Descartes’s rule of signs had said that no positive roots existed, you wouldn’t test any positives!)\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\">What do you have to multiply <i>x</i> in the divisor by to make it become 2<i>x</i><sup>4</sup> in the dividend? The quotient, 2<i>x</i><sup>3</sup>, goes above the 2<i>x</i><sup>4</sup> term.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\">Multiply this quotient by the divisor and write it under the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\">Subtract this line from the dividend: (2<i>x</i><sup>4</sup> – 9<i>x</i><sup>3</sup>) – (2<i>x</i><sup>4</sup> – 4<i>x</i><sup>3</sup>) = –5<i>x</i><sup>3</sup>. If you’ve done the job right, the subtraction of the first terms always produces 0.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\">Bring down the other terms of the dividend.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What do you have to multiply <i>x</i> by to make it –5<i>x</i><sup>3</sup>? Put the answer, –5<i>x</i><sup>2</sup>, above the –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>Multiply the –5<i>x</i><sup>2</sup> times the <i>x</i> – 2 to get –5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>. Write it under the remainder with the degrees lined up.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You now have (–5<i>x</i><sup>3</sup> – 21<i>x</i><sup>2</sup>) – (–5<i>x</i><sup>3</sup> + 10<i>x</i><sup>2</sup>) = –31<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +88<i>x</i> takes its place.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>What to multiply by to make <i>x</i> become –31<i>x</i><sup>2</sup>? The quotient –31<i>x</i> goes above –21<i>x</i><sup>2</sup>.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The value –31<i>x</i> times (<i>x</i> – 2) is –31<i>x</i><sup>2</sup> + 62<i>x;</i> write it under the remainder.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b> You now have (–31<i>x</i><sup>2</sup> + 88<i>x</i>) – (–31<i>x</i><sup>2</sup> + 62<i>x</i>), which is 26<i>x.</i></p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>B</u>ring down.</p>\r\n<p class=\"child-para\"><b></b>The +48 comes down.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>D</u>ivide.</p>\r\n<p class=\"child-para\"><b></b>The term 26<i>x</i> divided by <i>x</i> is 26. This answer goes on top.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>M</u>ultiply.</p>\r\n<p class=\"child-para\"><b></b>The constant 26 multiplied by (<i>x</i> – 2) is 26<i>x</i> – 52.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>ubtract.</p>\r\n<p class=\"child-para\"><b></b>You subtract (26<i>x</i> + 48) – (26<i>x</i> – 52) to get 100.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><u>S</u>top.</p>\r\n<p class=\"child-para\"><b></b>The remainder 100 has a degree that’s less than the divisor of <i>x</i> – 2.</p>\r\n</li>\r\n</ol>\r\nWow . . . now you know why they call it <i>long</i> division. You went through all that to find out that <i>x</i> – 2 isn’t a factor of the polynomial, which means that <i>x</i> = 2 isn’t a root.\r\n<p class=\"Remember\">If you divide by <i>c</i> and the remainder is 0, then the linear expression (<i>x</i> – <i>c</i>) is a factor and that <i>c</i> is a root. A remainder other than 0 implies that (<i>x</i> – <i>c</i>) isn’t a factor and that <i>c</i> isn’t a root.</p>","blurb":"","authors":[{"authorId":9703,"name":"Yang Kuang","slug":"yang-kuang","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9703"}},{"authorId":9704,"name":"Elleyne Kase","slug":"elleyne-kase","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9704"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to 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Whether you're an apprentice or a fully trained mathmagician, we have clear instruction to help you advance in the craft of math. Start with the basics and work up to calculus, plus everything in between. Yes, you do use this stuff in daily life.

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Basic Math AS and A-Level Maths For Dummies Cheat Sheet

Cheat Sheet / Updated 02-16-2023

Some of the most important things to remember in AS-level and A-level maths are the rules for differentiating and integrating expressions. This cheat sheet is a handy reference for what happens when you differentiate or integrate powers of x, trigonometric functions, exponentials or logarithms – as well as the rules you need for what to do when they’re combined!

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Trigonometry Trigonometry For Dummies Cheat Sheet

Cheat Sheet / Updated 02-09-2023

Trigonometry is the study of triangles, which contain angles, of course. Get to know some special rules for angles and various other important functions, definitions, and translations. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if you plan to study trig for very long.

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Statistics Statistics All-in-One For Dummies Cheat Sheet

Cheat Sheet / Updated 01-19-2023

Statistics involves a lot of data analysis, and analysis is built with math notation and formulas — but never fear, your cheat sheet is here to help you organize, understand, and remember the notation and formulas so that when it comes to putting them into practice or to the test, you’re ahead of the game!

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Algebra Algebra: How to Multiply and Divide Exponents

Article / Updated 12-21-2022

So, what is an exponent anyway? According to the Oxford dictionary, an exponent is defined as "a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression." Exponents are used in almost all levels of math, from algebra to calculus to physics. Here are two ways you can work with exponents when they show up in formulas and equations. How to multiply exponents You can multiply many exponential expressions together without having to change their form into the big or small numbers they represent. When multiplying exponents, the only requirement is that the bases of the exponential expressions have to be the same. So, you can multiply because the bases are not the same (although the exponents are). To multiply powers of the same base, add the exponents together: If there’s more than one base in an expression with powers, you can combine the numbers with the same bases, find the values, and then write them all together. For example, Here's an example with a number that has no exponent showing: When there’s no exponent showing, such as with y, you assume that the exponent is 1, so in the above example, you write How to divide exponents You can divide exponential expressions, leaving the answers as exponential expressions, as long as the bases are the same. To divide exponents (or powers) with the same base, subtract the exponents. Division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base. For example, Pretty easy, huh? Now wrap your brain around this: Any number to the power of zero equals 1, as long as the base number is not 0.

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Basic Math How to Calculate Percentages

Article / Updated 11-04-2022

Listen to the article:Download audio Whether you're leaving a tip at a restaurant or figuring out just how much those stylish shoes are on sale, you can't get away from percentages. While there are numerous percentage calculators online, it's helpful to be able to do some quick math in your head to calculate percentages without any digital assistance. What is percentage? The word percentage comes from the word percent. If you split the word percent into its root words, you see “per” and “cent.” Cent is an old European word with French, Latin, and Italian origins meaning “hundred." So, percent is translated directly to “per hundred.” If you have 87 percent, you literally have 87 per 100. If it snowed 13 times in the last 100 days, it snowed 13 percent of the time. How to find percentage The numbers that you will be converting into percentages can be given to you in two different formats: decimal and fraction. Decimal format is easier to calculate into a percentage. Converting a decimal to a percentage is as simple as multiplying it by 100. To convert .87 to a percent, simply multiply .87 by 100. .87 × 100=87, which gives us 87 percent. Percent is often abbreviated with the % symbol. You can present your answer as 87% or 87 percent — either way is acceptable. If you are given a fraction, convert it to a percentage by dividing the top number by the bottom number. If you are given 13/100, you would divide 13 by 100. 13 ÷ 100 = .13 Then, follow the steps above for converting a decimal to a percent. .13 × 100 = 13, thus giving you 13%. The more difficult task comes when you need to know a percentage when you are given numbers that don’t fit so neatly into 100. Most of the time, you will be given a percentage of a specific number. For example, you may know that 40 percent of your paycheck will go to taxes and you want to find out how much money that is. How to calculate percentage of a specific number This process is the reverse of what you did earlier. First convert the percentage number to a decimal. Then, you divide your percentage by 100. So, 40 percent would be 40 divided by 100. 40 ÷ 100 = .40 Next, once you have the decimal version of your percentage, simply multiply it by the given number (in this case, the amount of your paycheck). If your paycheck is $750, you would multiply 750 by .40. 750 × .40 = 300 Your answer would be 300. You are paying $300 in taxes. Let’s try another example. You need to save 25 percent of your paycheck for the next 6 months to pay for an upcoming vacation. If your paycheck is $1,500, how much should you save? Start by converting 25 percent to a decimal. 25 ÷ 100 = .25 Now, multiply the decimal by the amount of your paycheck, or 1500. 1500 × .25 = 375 This means you need to save $375 from each paycheck.

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Geometry Calculate the Volume of a Cylinder

Article / Updated 10-26-2022

The volume of an object is how much space the object takes up — or, if you were to drop the object into a full tub of water, how much water would overflow. To calculate the volume of a cylinder, you need to know its height and the area of its base. Because a cylinder is a flat-top figure (a solid with two congruent, parallel bases), the base can be either the top or bottom. If you know a cylinder's height and lateral area, but not its radius, you can use the formula for surface area to find the radius, and then calculate the volume from there. The lateral area of a cylinder is basically one rectangle rolled into a tube shape. Think of the lateral area of a cylinder as one rectangular paper towel that rolls exactly once around a paper towel roll. The base of this rectangle (you know, the part of the towel that wraps around the bottom of the roll) is the same as the circumference of the cylinder's base. And the height of the paper towel is the same as the height of the cylinder. Use this formula to calculate the volume of a cylinder Now for a cylinder problem: Here's a diagram to help you. To use the volume formula, you need the cylinder's height (which you know) and the area of its base. To get the area of the base, you need its radius. And to get the radius, you can use the surface area formula and solve for r: Remember that this "rectangle" is rolled around the cylinder and that the "rectangle's" base is the circumference of the cylinder's circular base. You fill in the equation as follows: Now set the equation equal to zero and factor: The radius can't be negative, so it's 5. Now you can finish with the volume formula: That does it.

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Statistics How to Use the T-table to Solve Statistics Problems

Article / Updated 10-26-2022

The t-table (for the t-distribution) is different from the z-table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various t-distributions, using the t-table, is a valuable statistics skill. How to use the t-table to find right-tail probabilities and p-values for hypothesis tests involving t: First, find the t-value for which you want the right-tail probability (call it t), and find the sample size (for example, n). Next, find the row corresponding to the degrees of freedom (df) for your problem (for example, n – 1). Go across that row to find the two t-values between which your t falls. For example, if your t is 1.60 and your n is 7, you look in the row for df = 7 – 1 = 6. Across that row you find your t lies between t-values 1.44 and 1.94. Then, go to the top of the columns containing the two t-values from Step 2. The right-tail (greater-than) probability for your t-value is somewhere between the two values at the top of these columns. For example, your t = 1.60 is between t-values 1.44 and 1.94 (df = 6); so the right tail probability for your t is between 0.10 (column heading for t = 1.44); and 0.05 (column heading for t = 1.94). The row near the bottom with Z in the df column gives right-tail (greater-than) probabilities from the Z-distribution. Use the t table to find t*-values (critical values) for a confidence interval involving t: Determine the confidence level you need (as a percentage). Determine the sample size (for example, n). Look at the bottom row of the table where the percentages are shown. Find your % confidence level there. Intersect this column with the row representing your degrees of freedom (df). This is the t-value you need for your confidence interval. For example, a 95% confidence interval with df=6 has t*=2.45. (Find 95% on the last line and go up to row 6.) Practice solving problems using the t-table sample questions below For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results? Answer: df = 17 The study involving one population and a sample size of 18 has n – 1 = 18 – 1 = 17 degrees of freedom. For a study involving a paired design with a total of 44 observations, with the results assuming a t-distribution, what row of the table will you use to find the probability affiliated with the study results? Answer: df = 21 A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: n – 1 = 22 – 1 = 21. A t-value of 2.35, from a t-distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table? Answer: 0.025 and 0.01 Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01. Hence, the upper-tail probability for a t-value of 2.35 must lie between 0.025 and 0.01.

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Pre-Algebra 10 Alternative Numeral and Number Systems

Article / Updated 10-24-2022

The distinction between numbers and numerals is subtle but important. A number is an idea that expresses how much or how many. A numeral is a written symbol that expresses a number. Here are ten ways to represent numbers that differ from the Hindu-Arabic (decimal) system. Tally marks Numbers are abstractions that stand for real things. The first known numbers came into being with the rise of trading and commerce — people needed to keep track of commodities such as animals, harvested crops, or tools. At first, traders used clay or stone tokens to help simplify the job of counting. Over time, tally marks scratched either in bone or on clay took the place of tokens. Bundled tally marks As early humans grew more comfortable letting tally marks stand for real-world objects, the next development in numbers was probably tally marks scratched in bundles of 5 (fingers on one hand), 10 (fingers on both hands), or 20 (fingers and toes). Bundling provided a simple way to count larger numbers more easily. Of course, this system is much easier to read than non-bundled scratches — you can easily multiply or count by fives to get the total. Even today, people keep track of points in games using bundles such as these. Egyptian numerals Ancient Egyptian numerals are among the oldest number systems still in use today. Egyptian numerals use seven symbols. Egyptian Numerals Number Symbol 1 Stroke 10 Yoke 100 Coil of rope 1,000 Lotus 10,000 Finger 100,000 Frog 1,000,000 Man with raised hands Numbers are formed by accumulating enough of the symbols that you need. For example, 7 = 7 strokes 24 = 2 yokes, 4 strokes 1,536 = 1 lotus, 5 coils of rope, 3 yokes, 6 strokes Babylonian numerals Babylonian numerals, which came into being about 4,000 years ago, use two symbols: 1 = Y 10 = < For numbers less than 60, numbers are formed by accumulating enough of the symbols you need. For example, 6 = YYYYYY 34 = << For numbers 60 and beyond, Babylonian numerals use place value based on the number 60. 61 = Y Y (one 60 and one 1) 124 = YY YYYY (two 60s and four 1s) 611 = < (ten 60s and eleven 1s) Ancient Greek numerals Ancient Greek numerals were based on the Greek letters. The numbers from 1 to 999 were formed using the symbols shown: Roman numerals Although Roman numerals are over 2,000 years old, people still use them today, either decoratively (for example, on clocks, cornerstones, and Super Bowl memorabilia) or when numerals distinct from decimal numbers are needed (for example, in outlines). Roman numerals use seven symbols, all of which are capital letters in the Latin alphabet (which pretty much happens to be the English alphabet as well): I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1,000 Mayan numerals Mayan numerals developed in South America during roughly the same period that Roman numerals developed in Europe. Mayan numerals use two symbols: dots and horizontal bars. A bar is equal to 5, and a dot is equal to 1. Numbers from 1 to 19 are formed by accumulating dots and bars. For example, 3 = 3 dots 7 = 2 dots over 1 bar 19 = 4 dots over 3 bars Numbers from 20 to 399 are formed using these same combinations, but raised up to indicate place value. For example, 21 = raised 1 dot, 1 dot (one 20 + one 1) 399 = raised 4 dots over 3 bars, 4 dots over 3 bars (nineteen 20s + three 5s + four 1s) Base-2 (binary) numbers Binary numbers use only two symbols: 0 and 1. This simplicity makes binary numbers useful as the number system that computers use for data storage and computation. Like the decimal system you're most familiar with, binary numbers use place value. Unlike the decimal system, binary place value is based not on powers of ten (1, 10, 100, 1,000, and so forth) but on powers of two (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and so on), as seen here: Binary Place Values 512s 256s 128s 64s 32s 16s 8s 4s 2s 1s Base-16 (hexadecimal) numbers The computer's first language is binary numbers. But in practice, humans find binary numbers of any significant length virtually undecipherable. Hexadecimal numbers, however, are readable to humans and still easily translated into binary numbers, so computer programmers use hexadecimal numbers as a sort of common language when interfacing with computers at the deepest level, the level of hardware and software design. The hexadecimal number system uses all ten digits 0 through 9 from the decimal system. Additionally, it uses six more symbols: A = 10 B = 11 C = 12 D = 13 E = 14 F = 15 Hexadecimal is a place-value system based on powers of 16. Hexadecimal Place Values 1,048,576s 65,536s 4,096s 256s 16s 1s As you can see, each number in the table is exactly 16 times the number to its immediate right. Prime-based numbers One wacky way to represent numbers unlike any of the others is prime-based numbers. Prime-based numbers are similar to decimal, binary, and hexadecimal numbers in that they use place value to determine the value of digits. But unlike these other number systems, prime-based numbers are based not on addition but on multiplication. Prime-Based Place Values 31s 29s 23s 19s 17s 13s 11s 7s 5s 3s 2s You can use the table to find the decimal value of a prime-based number.

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

Article / Updated 10-06-2022

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

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