{"id":7427,"date":"2014-09-17T17:59:03","date_gmt":"2014-09-17T17:59:03","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=7427"},"modified":"2014-12-02T08:25:26","modified_gmt":"2014-12-02T08:25:26","slug":"the-foil-method-of-factoring-polynomials-when-a-is-not-equal-to-1","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/the-foil-method-of-factoring-polynomials-when-a-is-not-equal-to-1\/","title":{"rendered":"The FOIL Method of Factoring Polynomials when A Is Not Equal to 1"},"content":{"rendered":"

Overview<\/h3>\n

When the a term has a coefficient of 1, polynomials can be factored into the form (a +b) (a +c) without worrying about its effect. However, if that a term has a coefficient not equal to 1, the coefficient must be considered when the polynomial is factored.<\/p>\n

Review of FOIL<\/h3>\n

Suppose (x + 2) (x + 3) is multiplied. The First terms, x \u2219x, become x2<\/sup>. The Outer terms, 3x, are next, then the inner terms, 2x. The last terms 2\u22193, equal 6. When the polynomial is put in order, it follows the FOIL pattern of x2<\/sup> + 2x + 3x + 6, or x2<\/sup> + 5x + 6. (It follows the pattern of x2<\/sup> + bx + c.) Because x2<\/sup> is equal to 1\u2219 x2<\/sup> in that case, the FOIL method can still be used. It just takes more attention to factor the expression.<\/p>\n

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

Common Factors<\/h3>\n

Factor out the largest common factor if that a term is not equal to 1 before using FOIL to do the rest. Suppose the polynomial is 5x2<\/sup> + 15x + 125. The largest common factor is 5 in all the terms. Before using FOIL, the trinomial can be partially factored as 5(x2<\/sup> + 10x + 25). Then it can be factored the rest of the way as 5(x + 5) (x + 5). Suppose that the trinomial is 24x2<\/sup> + 76x + 40. The common factor of 4 can be factored out to leave 4(6x2<\/sup> + 19x + 10).<\/p>\n

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

UnFOILing the FOIL<\/h3>\n

In the above example, 6x2<\/sup> + 19x + 10, the a term is equal to 6. Therefore, the two First terms will have a product of 6x2<\/sup>. They could be x and 6x or 2x and 3x, as x\u2219 6x = 6x2<\/sup>, and 2x\u22193x is also 6x2<\/sup>. The two Last terms will have a product of 10, so they could be 5\u22192; 2\u22195; 1\u221910; or 10\u22191. Then use trial and error to determine which Outer and Inner terms will have a sum of 19, the b term. Suppose the first combination is (x +5) (6x + 2). Using FOIL, the multiplication will result in 6x2<\/sup> + 2x +30x +10. The 6x2<\/sup> is correct, and the 10, but the middle term, 32x, is too large. What about (x +2) (6x +5)? The only terms to try are 5x + 12x, which is 17x, too small. Similarly, (x + 1) (6x + 10) results in a middle term of 16x, smaller still; and (x + 10) (6x + 1); 61x. Next, (2x + 1) (3x + 10). The sum of the middle terms is 23x, still too large; and (2x + 10) (3x + 1); 36x. Next, (2x + 2) (3x + 5), 16x again. That leaves (2x + 5) (3x + 2); 19x.<\/p>\n

Reading the Signs<\/h3>\n

Whatever the value is of a, the sign within the factored terms matter. Suppose the trinomial is of the form ax2<\/sup> +bx +c. It will factor as two additions in (x + p) (x + q). If the trinomial is in the form ax2<\/sup> \u2013bx +c, it will factor as two subtractions (x \u2013 p) (x – q). If the last term is negative, as in ax2<\/sup> +bx \u2013c or ax2<\/sup> \u2013bx \u2013c, it will factor as either (x + p) (x – q) or (x – p) (x + q). If the middle term is positive, then p is less than q, and (x + p) (x \u2013 q) is the correct choice. If the middle term is negative, then q is less than p and (x \u2013 p) (x + q) is the correct choice.<\/p>\n

Figure 3: The direction of the signs matter.<\/p>\n

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

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Overview When the a term has a coefficient of 1, polynomials can be factored into the form (a +b) (a +c) without worrying about its effect. However, if that a term has a coefficient not equal to 1, the coefficient must be considered when the polynomial is factored. Review of FOIL Suppose (x + 2) […]<\/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":[2775,2776],"class_list":["post-7427","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-coefficients-of-polynomials","tag-factoring-by-using-foil"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7427","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=7427"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7427\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=7427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=7427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=7427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}