{"id":7454,"date":"2014-10-03T17:47:53","date_gmt":"2014-10-03T17:47:53","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=7454"},"modified":"2014-12-02T08:25:25","modified_gmt":"2014-12-02T08:25:25","slug":"math-review-of-special-factoring-formulas-and-a-general-review-of-factoring","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/math-review-of-special-factoring-formulas-and-a-general-review-of-factoring\/","title":{"rendered":"Math Review of Special Factoring Formulas and a General Review of Factoring"},"content":{"rendered":"

Overview<\/h3>\n

Some special factoring formulas include the difference of two squares, the sum of two cubes, and the difference of two cubes. If there are three terms or more in the polynomial, students can use strategies such as finding common factors and factoring by grouping.<\/p>\n

Extending the Difference of Two Squares<\/h3>\n

The difference of two squares [(a + b)(a – b)] is a common pattern with binomials involving variables to the second power. However, the concept can also be applied to exponents higher than x2<\/sup>. Any even power (such as x2<\/sup>, x4<\/sup>, x6<\/sup>, and so on) can be factored into squares evenly. For example, 16x4<\/sup> can be rewritten as (4x2<\/sup>)2<\/sup> and 49y6<\/sup> can be rewritten as (7y3<\/sup>)2<\/sup>. The expression 16x4<\/sup> \u2013 49y6<\/sup> is then factored as (4x2<\/sup> – 7y3<\/sup>)(4x2<\/sup> + 7y3<\/sup>).<\/p>\n

Sum of Two Cubes<\/h3>\n

The product of (a + b)(a2<\/sup> \u2013 ab + b2<\/sup>) can be evaluated using FOIL as a3<\/sup> \u2013a2<\/sup>b + ab2<\/sup> +a2<\/sup>b \u2013 ab2<\/sup> + b3<\/sup>. That simplifies to a3<\/sup> + b3<\/sup>. Suppose that the binomial that needs to be factored is 27x3<\/sup> + 8. That expression will factor as (3x + 2)(9x2<\/sup> – 6x + 4).<\/p>\n

Figure 1: Factoring the sum of two cubes and the difference of two cubes.<\/p>\n

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

Difference of Two Cubes<\/h3>\n

The product of (a – b)(a2<\/sup> + ab + b2<\/sup>) can also be evaluated and simplified to a3<\/sup> \u2013 b3<\/sup>. The easiest way to remember the direction of the signs when factoring the sum or difference between two cubes is to use the acronym SOAP. The sign between the terms of the binomial factor is in the same direction in both the sum of the cubes and the (a + b) factor. (If the difference of cubes is the issue, the sign in a3<\/sup> \u2013 b3<\/sup> and a \u2013 b is negative.) The sign is opposite between the a2<\/sup> term and the ab term, such that if it is the sum of cubes the sign between a2<\/sup> and ab is negative, and if it is the difference in cubes, the sign between the a2<\/sup> and ab term is positive. The sign between the ab term and the constant is always positive.<\/p>\n

Figure 2: Using the acronym SOAP to remember the direction of the signs.<\/p>\n

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

Extending Factoring Strategies<\/h3>\n

The first step in factoring a polynomial is always to factor out anything that is common to every term in the polynomial. Suppose that the polynomial to be factored is 3x2<\/sup> + 6x + 9. The first step in factoring would be to remove the common factor of 3 from all the terms as 3(x2<\/sup> + 2x + 3). Next, check to see if it follows any of the special factoring forms. It can be factored by grouping or another method.<\/p>\n

Figure 3: Following the general steps to factor a polynomial.<\/p>\n

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

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Overview Some special factoring formulas include the difference of two squares, the sum of two cubes, and the difference of two cubes. If there are three terms or more in the polynomial, students can use strategies such as finding common factors and factoring by grouping. Extending the Difference of Two Squares The difference of two […]<\/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":[319,2751,2725,2750],"class_list":["post-7454","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-common-factors","tag-difference-of-cubes","tag-difference-of-squares","tag-sum-of-cubes"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7454","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=7454"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7454\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=7454"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=7454"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=7454"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}