{"id":6797,"date":"2014-04-29T17:04:27","date_gmt":"2014-04-29T17:04:27","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=6797"},"modified":"2014-12-02T08:25:30","modified_gmt":"2014-12-02T08:25:30","slug":"math-introduction-to-factoring-a-perfect-square-trinomial","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/math-introduction-to-factoring-a-perfect-square-trinomial\/","title":{"rendered":"Math Introduction to Factoring a Perfect Square Trinomial"},"content":{"rendered":"

Overview:<\/strong><\/p>\n

A perfect square trinomial is the square of a binomial.\u00a0 It follows a pattern when it is factored, so that the first and last terms are perfect squares of monomials and the middle term is twice their product.\u00a0 If the pattern does not fit for a particular trinomial, it is not a perfect square trinomial.<\/p>\n

How Is A Binomial Squared?<\/strong><\/p>\n

In order to multiply two binomials (a + b)(c + d), the product is ac + bc + cb + bd, or in other words, ac + 2bc + bd.\u00a0 The process for squaring a binomial is the same.\u00a0 Suppose the binomial is (x + 3).\u00a0 Squaring that binomial, (in other words (x + 3)2<\/sup>), is the same thing as saying (x + 3)(x + 3).\u00a0 Following the pattern, multiplying x\u2219x, gives x2<\/sup>, while multiplying 3x and adding x3 equals 6x and multiplying 3\u22193 equals 9.\u00a0 The entire equation is x2<\/sup> + 6x + 9.<\/p>\n

What if the Binomial Has a Minus Sign?<\/strong><\/p>\n

Suppose the binomial were (x-4) instead.\u00a0 Squaring the binomial (x-4) is the same thing as multiplying (x-4)(x-4).\u00a0 Following the FOIL pattern, (x-4)(x-4) is the same thing as x2<\/sup> – 4x \u2013 x4 + 49.\u00a0 In other words, (x – 4)2<\/sup> equals x2 <\/sup>-8x + 49.\u00a0 The process is similar as to (a-b)(c-d), with a product of ac \u2013 bc \u2013 cb + bd.\u00a0 Multiplying two negatives, -b \u2219-d, changes the sign to a positive number.<\/p>\n

What Is the Pattern for a Perfect Square Trinomial?<\/strong><\/p>\n

The square of a binomial (x +a) is the same thing as multiplying (x + a)(x + a), or x2<\/sup> + 2ax + a2<\/sup>.\u00a0 The square of a binomial (x-a) is the same thing as multiplying (x-a)(x-a) or x2<\/sup>– 2ax + a2<\/sup>.\u00a0 Therefore, a trinomial that follows the pattern x2<\/sup> \u00b1 2ax + a2<\/sup> is the square of a binomial.<\/p>\n

Testing a Trinomial<\/strong><\/p>\n

Suppose the trinomial is 4x2<\/sup> + 20x + 25.\u00a0 The square root of 4x2<\/sup> is 2x and the square root of 25 is 5.\u00a0 The middle term 20x can be factored as 2\u22192x\u22195.\u00a0 It follows the pattern, and can be factored as 2x + 5, because (2x + 5)2<\/sup> equals 4x2<\/sup> + 20x + 25. Similarly, if the trinomial is x2<\/sup> -16x + 64, the square root of x2<\/sup> is x and the square root of 64 is 8. The number 64 also has a negative square root of -8, because -8 \u2219-8 also equals 64. The middle term -16x can be factored as 2\u2219-8\u2219x.\u00a0 It also follows the pattern, and can be factored as x-8, because (x-8)2<\/sup> equals x2<\/sup> -16x + 64.\u00a0 Suppose the trinomial were 36x2<\/sup> +23x + 81.\u00a0 Is it a perfect square trinomial?\u00a0 The square root of the monomial 36x2<\/sup> is 6x and the square root of 81 is 9, but the middle term 23x is not equal to 2\u22196x\u22199, or 108x.<\/p>\n

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Overview: A perfect square trinomial is the square of a binomial.\u00a0 It follows a pattern when it is factored, so that the first and last terms are perfect squares of monomials and the middle term is twice their product.\u00a0 If the pattern does not fit for a particular trinomial, it is not a perfect square […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[2635,2634],"class_list":["post-6797","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-perfect-square-trinomial","tag-squaring-a-binomial"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/6797","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=6797"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/6797\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=6797"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=6797"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=6797"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}