{"id":5673,"date":"2013-04-09T15:46:48","date_gmt":"2013-04-09T15:46:48","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=5673"},"modified":"2014-12-02T08:27:03","modified_gmt":"2014-12-02T08:27:03","slug":"factoring-polynomials","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/factoring-polynomials\/","title":{"rendered":"Factoring Polynomials"},"content":{"rendered":"

Overview:\u00a0 Factoring Polynomials<\/strong>
\nIn order to factor polynomials, it is important to find the greatest common factors and use the distributive property.\u00a0 Use the integral coefficients to rewrite the polynomial and find the factors.\u00a0 The difference between finding prime factors of a real number and prime polynomials is that there are variables involved.\u00a0 Multiplying back the number sentences is a way to check work.<\/p>\n

Finding the Greatest Common Factors<\/strong>
\nWhen finding the greatest common factors of a polynomial, look at the numerical coefficients first to see what their greatest common factors are.\u00a0 For example, 3x +3y can be written as 3(x +y), because both 3x and 3y have the factor 3 in common.\u00a0 Similarly, in the sentence 7x3<\/sup> -21x,\u00a0 both 7 and 21 have the factor 7 in common, and each term of the polynomial also has the factor x in common.\u00a0 Factoring for 7x\u00a0 involves division by 7x, so that 7x3<\/sup>\/7x equals x2<\/sup> and 21x\/7x equals 3.<\/p>\n

Using the Distributive Property and Grouping Like Terms<\/strong>
\nIn the example 7x3<\/sup>-21x, another mathematical property is important for factoring.\u00a0 The distributive property states that for any a, b, and c, a(b + c) = ab + ac, and a(b-c) = ab-ac.\u00a0 Therefore, 7x3<\/sup>-21x will equal 7x(x2<\/sup>-3) because 7x(x2<\/sup>) -7x(3) = 7x3<\/sup>-21x.\u00a0 When factoring ax +cy +xy + ac, group the x terms and the c terms together to make it easier to factor common terms, so that ax + cy +xy +ac = (ax + xy) + (cy +ac) = x (a +y) + c(y +a), and (a +y ) = ( y +a) .\u00a0 Therefore,( x +c) +(y +a) = ax +xy + ac + cy.<\/p>\n

Factoring the Difference of Two Squares<\/strong>
\nThis is another instance where algebra and geometry work together.\u00a0 Suppose the problem is to find the difference between the area covered by a large square with sides a\u00a0 and a small square with sides b.\u00a0 The area that is left will be covered by a2<\/sup> -b2<\/sup>, or(a -b) (a +b).\u00a0 Note that a negative times a positive gives a negative product.\u00a0 For example, x2<\/sup>-9 is the difference of two squares, as the square root of x2<\/sup> is x and the square root of 9 is 3.\u00a0 Therefore x2<\/sup>-9 can be factored as (x +3)(x-3).<\/p>\n

Factoring Two Squares with a Coefficient<\/strong>
\nIn the example of x2<\/sup>-9 = (x +3)(x-3), the x term did not have a coefficient.\u00a0 However, if the variable does have a coefficient, or if the second square has an additional variable, those can also be factored.\u00a0 For example, 25x2<\/sup>– 81 is also the difference of two squares:\u00a0 (5x – 9)(5x +9), as 25x2<\/sup> is equal to (5x)(5x).\u00a0 If the problem were 16x2<\/sup>-36y2<\/sup>, that would factor as (4x – 6y)(4x +6y)<\/p>\n

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Overview:\u00a0 Factoring Polynomials In order to factor polynomials, it is important to find the greatest common factors and use the distributive property.\u00a0 Use the integral coefficients to rewrite the polynomial and find the factors.\u00a0 The difference between finding prime factors of a real number and prime polynomials is that there are variables involved.\u00a0 Multiplying back […]<\/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":[496,519,645,751,1360],"class_list":["post-5673","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-difference-of-two-squares","tag-distributive-property","tag-factoring","tag-greatest-common-factor","tag-polynomials"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5673","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=5673"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5673\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=5673"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=5673"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=5673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}