{"id":7448,"date":"2015-01-08T22:49:12","date_gmt":"2015-01-08T22:49:12","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=7448"},"modified":"2015-01-08T22:49:12","modified_gmt":"2015-01-08T22:49:12","slug":"math-review-of-factoring-trinomial-squares-and-differences-of-squares","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/math-review-of-factoring-trinomial-squares-and-differences-of-squares\/","title":{"rendered":"Math Review of Factoring Trinomial Squares and Differences of Squares"},"content":{"rendered":"

Overview<\/h3>\n

Trinomial squares are also known as perfect square trinomials, and are the squares of binomial expressions. They factor as (a + b)(a + b) or (a – b)(a – b) where a and b are real numbers. Forms such as (a + b)(a -b) are special products that are also called the difference of squares.<\/p>\n

Trinomial Squares<\/h3>\n

When a squared binomial such as (a + b)(a + b) is multiplied using FOIL, the values of a and b follow a specific pattern. Recall that the first term is a2<\/sup>, the outside and inside terms are ab + ba, and the last term is b2<\/sup>. If a polynomial follows the form ax2<\/sup> + bx + c, a and c are perfect squares, and the b coefficient is twice the sum of ac, it is a perfect trinomial square. Suppose the polynomial is 36x2<\/sup> + 60x + 25. 36x2<\/sup> is a perfect square of 6x, and 25 is a perfect square of 5. The inner and outer terms are 30x + 30x or 60x. That polynomial is (6x + 5)2<\/sup>.<\/p>\n

Figure 1: The perfect square\u00a0trinomial follows a specific pattern.<\/p>\n

<\/a><\/p>\n

Negatives<\/h3>\n

What if the squared binomial is (a – b)(a – b)? When it is multiplied using FOIL, a2<\/sup> is a perfect square, and so is b2<\/sup>. The sign of 2ab is negative, because it is the sum of two negative products. Suppose the polynomial is 100x2<\/sup> \u2013 80x + 16. The square root of 100x2<\/sup> is 10x, and the square root of 16 is -4. The product of 10 and -4 is -40, and twice -40 is -80. That polynomial is (10x – 4)2<\/sup>.<\/p>\n

Figure 2: An example when the middle term is negative.<\/p>\n

<\/a><\/p>\n

Other Patterns<\/h3>\n

If the trinomial follows the form \u2013ax2<\/sup> +bx +c or ax2<\/sup> \u2013 bx \u2013 c or ax2<\/sup> + bx \u2013 c, it does not follow the squared trinomial pattern. The coefficient of a squared term cannot be negative even if the term is a perfect square. When a negative is multiplied by another negative, the product is positive. Similarly, if the constant c is not a perfect square, the trinomial does not follow the squared trinomial pattern.<\/p>\n

Differences of Squares<\/h3>\n

The last special pattern to consider is (a + b)(a – b). Since multiplication is commutative, it is also the same as (a – b)(a + b). It is called the difference of squares. When (a + b)(a – b) is multiplied using FOIL, the first term is a2<\/sup>, and the last term is b2<\/sup>. The outside term is \u2013ab and the inside term is ab, which adds up as zero. They cancel each other out. Suppose a polynomial is 144x2<\/sup> + 81. The square root of 144x2<\/sup> is 12x and the square root of 81 is 9. Following the pattern, the factoring is (12x + 9)(12x – 9).<\/p>\n

Figure 3: An example of the form (a + b)(a – b).<\/p>\n

<\/a><\/p>\n

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Overview Trinomial squares are also known as perfect square trinomials, and are the squares of binomial expressions. They factor as (a + b)(a + b) or (a – b)(a – b) where a and b are real numbers. Forms such as (a + b)(a -b) are special products that are also called the difference of […]<\/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":[2725,2635],"class_list":["post-7448","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-difference-of-squares","tag-perfect-square-trinomial"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7448","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=7448"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7448\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=7448"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=7448"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=7448"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}