{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T15:11:44+00:00","modifiedTime":"2016-03-26T15:11:44+00:00","timestamp":"2022-09-14T18:05:18+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Pre-Calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"How to Factor a Perfect Square","strippedTitle":"how to factor a perfect square","slug":"how-to-factor-a-perfect-square","canonicalUrl":"","seo":{"metaDescription":"FOIL stands for multiply the first, outside, inside, and last terms together. When you FOIL a binomial times itself, the product is called a perfect square. For","noIndex":0,"noFollow":0},"content":"<p>FOIL stands for multiply the <i>first, outside, inside,</i> and <i>last</i> terms together. When you FOIL a binomial times itself, the product is called a <i>perfect square.</i> For example, (<i>a</i> + <i>b</i>)<sup>2</sup> gives you the perfect-square trinomial <i>a</i><sup>2</sup> + 2<i>ab</i> + <i>b</i><sup>2</sup>. Because a perfect-square trinomial is still a trinomial, you follow the steps in the backward FOIL method of factoring. However,<i> </i>you must account for one extra step at the very end where you express the answer as a binomial squared.</p>\n<p>For example, to factor the polynomial 4<i>x</i><sup>2</sup> – 12<i>x </i>+ 9, follow these steps: </p>\n<ol class=\"level-one\">\n <li><p class=\"first-para\">Multiply the quadratic term and the constant term.</p>\n<p class=\"child-para\">The product of the quadratic term 4<i>x</i><sup>2</sup> and the constant 9 is 36<i>x</i><sup>2</sup>, so that made your job easy.</p>\n </li>\n <li><p class=\"first-para\">Write down all the factors of the result that result in pairs in which each term in the pair has an <i>x</i>.</p>\n<p class=\"child-para\">Following are the factors of 36<i>x</i><sup>2</sup> in pairs:</p>\n <ul class=\"level-two\">\n <li><p class=\"first-para\">1<i>x </i>and 36<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–1<i>x</i> and –36<i>x</i></p>\n </li>\n <li><p class=\"first-para\">2<i>x</i> and 18<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–2<i>x</i> and –18<i>x</i></p>\n </li>\n <li><p class=\"first-para\">3<i>x</i> and 12<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–3<i>x</i> and –12<i>x</i></p>\n </li>\n <li><p class=\"first-para\">4<i>x</i> and 9<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–4<i>x</i> and –9<i>x</i></p>\n </li>\n <li><p class=\"first-para\">6<i>x</i> and 6<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–6<i>x</i> and –6<i>x</i></p>\n </li>\n </ul>\n<p class=\"child-para\">If you think ahead to the next step, you can skip writing out the positive factors, because they produce only <i>x</i> terms with a positive coefficient.</p>\n </li>\n <li><p class=\"first-para\">From this list, find the pair that adds to produce the coefficient of the linear term.</p>\n<p class=\"child-para\">You want to get a sum of –12<i>x</i> in this case. The only way to do that is to use –6<i>x</i> and –6<i>x.</i></p>\n </li>\n <li><p class=\"first-para\">Break up the linear term into two terms, using the terms from Step 3.</p>\n<p class=\"child-para\">You now get 4<i>x</i><sup>2</sup> – 6<i>x </i>– 6<i>x</i> + 9.</p>\n </li>\n <li><p class=\"first-para\">Group the four terms into two sets of two.</p>\n<p class=\"child-para\">Remember to include the plus sign between the two groups, resulting in (4<i>x</i><sup>2</sup> – 6<i>x</i>) + (–6<i>x</i> + 9).</p>\n </li>\n <li><p class=\"first-para\">Find the Greatest Common Factor (GCF) for each set and factor it out.</p>\n<p class=\"child-para\">The GCF of the first two terms is 2<i>x,</i> and the GCF of the next two terms is –3; when you factor them out, you get 2<i>x</i>(2<i>x</i> – 3) – 3(2<i>x</i> – 3).</p>\n </li>\n <li><p class=\"first-para\">Find the GCF of the two new terms.</p>\n<p class=\"child-para\">This time the GCF is (2<i>x</i> – 3); when you factor it out, you get (2<i>x</i> – 3)(2<i>x</i> – 3). Aha! That’s a binomial times itself, which means you have one extra step.</p>\n </li>\n <li><p class=\"first-para\">Express the resulting product as a binomial squared.</p>\n<p class=\"child-para\">This step is easy: (2<i>x</i> – 3)<sup>2</sup>.</p>\n </li>\n</ol>","description":"<p>FOIL stands for multiply the <i>first, outside, inside,</i> and <i>last</i> terms together. When you FOIL a binomial times itself, the product is called a <i>perfect square.</i> For example, (<i>a</i> + <i>b</i>)<sup>2</sup> gives you the perfect-square trinomial <i>a</i><sup>2</sup> + 2<i>ab</i> + <i>b</i><sup>2</sup>. Because a perfect-square trinomial is still a trinomial, you follow the steps in the backward FOIL method of factoring. However,<i> </i>you must account for one extra step at the very end where you express the answer as a binomial squared.</p>\n<p>For example, to factor the polynomial 4<i>x</i><sup>2</sup> – 12<i>x </i>+ 9, follow these steps: </p>\n<ol class=\"level-one\">\n <li><p class=\"first-para\">Multiply the quadratic term and the constant term.</p>\n<p class=\"child-para\">The product of the quadratic term 4<i>x</i><sup>2</sup> and the constant 9 is 36<i>x</i><sup>2</sup>, so that made your job easy.</p>\n </li>\n <li><p class=\"first-para\">Write down all the factors of the result that result in pairs in which each term in the pair has an <i>x</i>.</p>\n<p class=\"child-para\">Following are the factors of 36<i>x</i><sup>2</sup> in pairs:</p>\n <ul class=\"level-two\">\n <li><p class=\"first-para\">1<i>x </i>and 36<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–1<i>x</i> and –36<i>x</i></p>\n </li>\n <li><p class=\"first-para\">2<i>x</i> and 18<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–2<i>x</i> and –18<i>x</i></p>\n </li>\n <li><p class=\"first-para\">3<i>x</i> and 12<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–3<i>x</i> and –12<i>x</i></p>\n </li>\n <li><p class=\"first-para\">4<i>x</i> and 9<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–4<i>x</i> and –9<i>x</i></p>\n </li>\n <li><p class=\"first-para\">6<i>x</i> and 6<i>x</i></p>\n </li>\n <li><p class=\"first-para\">–6<i>x</i> and –6<i>x</i></p>\n </li>\n </ul>\n<p class=\"child-para\">If you think ahead to the next step, you can skip writing out the positive factors, because they produce only <i>x</i> terms with a positive coefficient.</p>\n </li>\n <li><p class=\"first-para\">From this list, find the pair that adds to produce the coefficient of the linear term.</p>\n<p class=\"child-para\">You want to get a sum of –12<i>x</i> in this case. The only way to do that is to use –6<i>x</i> and –6<i>x.</i></p>\n </li>\n <li><p class=\"first-para\">Break up the linear term into two terms, using the terms from Step 3.</p>\n<p class=\"child-para\">You now get 4<i>x</i><sup>2</sup> – 6<i>x </i>– 6<i>x</i> + 9.</p>\n </li>\n <li><p class=\"first-para\">Group the four terms into two sets of two.</p>\n<p class=\"child-para\">Remember to include the plus sign between the two groups, resulting in (4<i>x</i><sup>2</sup> – 6<i>x</i>) + (–6<i>x</i> + 9).</p>\n </li>\n <li><p class=\"first-para\">Find the Greatest Common Factor (GCF) for each set and factor it out.</p>\n<p class=\"child-para\">The GCF of the first two terms is 2<i>x,</i> and the GCF of the next two terms is –3; when you factor them out, you get 2<i>x</i>(2<i>x</i> – 3) – 3(2<i>x</i> – 3).</p>\n </li>\n <li><p class=\"first-para\">Find the GCF of the two new terms.</p>\n<p class=\"child-para\">This time the GCF is (2<i>x</i> – 3); when you factor it out, you get (2<i>x</i> – 3)(2<i>x</i> – 3). Aha! That’s a binomial times itself, which means you have one extra step.</p>\n </li>\n <li><p class=\"first-para\">Express the resulting product as a binomial squared.</p>\n<p class=\"child-para\">This step is easy: (2<i>x</i> – 3)<sup>2</sup>.</p>\n </li>\n</ol>","blurb":"","authors":[{"authorId":9703,"name":"Yang Kuang","slug":"yang-kuang","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9703"}},{"authorId":9704,"name":"Elleyne Kase","slug":"elleyne-kase","description":"","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9704"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"https://dummies-api.dummies.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":true,"relatedBook":{"bookId":282498,"slug":"pre-calculus-for-dummies-1001-practice-problems","isbn":"9781119883623","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"https://www.amazon.com/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119883628-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119883628/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/covers/9781119883623.jpg","width":250,"height":350},"title":"Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"\n <p><p><b><b data-author-id=\"8985\">Mary Jane Sterling</b></b> was a professor of mathematics at Bradley University. She taught for more than 35 years and continues to teach via distance learning. Sterling is the author of many Dummies math titles, including <i>Pre-Calculus For Dummies</i>. She is a graduate of the University of New Hampshire with a master’s degree in mathematics education.</p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> was a professor of mathematics at Bradley University. She taught for more than 35 years and continues to teach via distance learning. Sterling is the author of many Dummies math titles, including <i>Pre-Calculus For Dummies</i>. She is a graduate of the University of New Hampshire with a master’s degree in mathematics education. 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How to Factor a Perfect Square

By: Yang Kuang and Elleyne Kase and

Updated: 03-26-2016

Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)

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FOIL stands for multiply the first, outside, inside, and last terms together. When you FOIL a binomial times itself, the product is called a perfect square. For example, (a + b)² gives you the perfect-square trinomial a² + 2ab + b². Because a perfect-square trinomial is still a trinomial, you follow the steps in the backward FOIL method of factoring. However, you must account for one extra step at the very end where you express the answer as a binomial squared.

For example, to factor the polynomial 4x² – 12x + 9, follow these steps:

Multiply the quadratic term and the constant term.

The product of the quadratic term 4x² and the constant 9 is 36x², so that made your job easy.
Write down all the factors of the result that result in pairs in which each term in the pair has an x.

Following are the factors of 36x² in pairs:
- 1x and 36x
- –1x and –36x
- 2x and 18x
- –2x and –18x
- 3x and 12x
- –3x and –12x
- 4x and 9x
- –4x and –9x
- 6x and 6x
- –6x and –6x
If you think ahead to the next step, you can skip writing out the positive factors, because they produce only x terms with a positive coefficient.
From this list, find the pair that adds to produce the coefficient of the linear term.

You want to get a sum of –12x in this case. The only way to do that is to use –6x and –6x.
Break up the linear term into two terms, using the terms from Step 3.

You now get 4x² – 6x – 6x + 9.
Group the four terms into two sets of two.

Remember to include the plus sign between the two groups, resulting in (4x² – 6x) + (–6x + 9).
Find the Greatest Common Factor (GCF) for each set and factor it out.

The GCF of the first two terms is 2x, and the GCF of the next two terms is –3; when you factor them out, you get 2x(2x – 3) – 3(2x – 3).
Find the GCF of the two new terms.

This time the GCF is (2x – 3); when you factor it out, you get (2x – 3)(2x – 3). Aha! That’s a binomial times itself, which means you have one extra step.
Express the resulting product as a binomial squared.

This step is easy: (2x – 3)².

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