Figure 1: The sum of cubes follows the pattern x3<\/sup> + a3<\/sup>.<\/p>\n A difference of cubes is an expressions such as x3<\/sup> – a3<\/sup>, where both members of the expression are perfect cubes. Suppose the expression is 64x3<\/sup> \u2013 125. The monomial 64x3<\/sup> is a perfect cube of 4x, because (4x)3<\/sup> equals 64x3<\/sup>. Similarly, the constant 125 is a perfect cube of 5, because 53<\/sup> is 125. A perfect cube can be a negative real number, because a negative real number times a negative real number is positive, and a positive number times a negative number is a negative number.<\/p>\n Figure 2: The difference of cubes follows the pattern x3<\/sup> \u2013 a3<\/sup>.<\/p>\n The sum of cubes x3<\/sup> + a3<\/sup> follows a special pattern. One factor is (x + a), and the other factor is a quadratic polynomial that is already in simplest terms, (x2<\/sup> \u2013 ax + a2<\/sup>). Multiplying (x + a) (x2<\/sup> \u2013 ax + a2<\/sup>) is the same thing as adding x(x2<\/sup> – ax + a2<\/sup>) + a(x2<\/sup> \u2013 ax + a2<\/sup>). The first term is x3<\/sup> – ax2<\/sup> + a2<\/sup>x, and the second term is ax2<\/sup> \u2013 a2<\/sup>x +a3<\/sup>. Putting the terms together, the entire expression is x3<\/sup> \u2013 ax2<\/sup> + a2<\/sup>x \u2013a2<\/sup>x + a3<\/sup>. Simplified, the expression is the sum of cubes x3<\/sup> + a3<\/sup>. Suppose the expression is 8y3<\/sup> +27. Following the pattern, it can be factored as (2y + 3) (4y2<\/sup> – 6y + 9).<\/p>\n The difference of cubes x3<\/sup> \u2013 a3<\/sup> also follows a special pattern. One factor is (x – a), and the other factor is a similar quadratic polynomial to the sum of cubes, also already in simplest terms, (x2<\/sup> +ax +a2<\/sup>). Multiplying (x – a)(x2<\/sup> + ax + a2<\/sup>) is the same thing as adding x(x2<\/sup> + ax + a2<\/sup>) \u2013 a(x2<\/sup> + ax + a2<\/sup>). Simplified, the expression is the difference of cubes x3<\/sup> – a3<\/sup>. Suppose the expression is x3<\/sup> \u2013 216. Following the pattern, it can be factored as (x – 6)(x2<\/sup> + 6x + 36). That is the same thing as x(x2<\/sup> + 6x + 36) \u2013 6(x2<\/sup> +6x + 36). Putting the terms together, the entire expression is x3<\/sup> + 6x2 <\/sup> + 36x – 6x2<\/sup> \u2013 36x – 216.<\/p>\n Figure 3: The pattern for factoring the sum of two cubes or the difference of two cubes.<\/p>\n The acronym SOAP is an easy way to remember the sequence for either the sum or difference of cubes. If the expression to be factored is a sum of cubes x3<\/sup> +a3<\/sup>, the first factor (x + a) has the same sign as x3<\/sup> + a3<\/sup>. The first operation (x2<\/sup> – ax) in the second factor (x2<\/sup> \u2013 ax + b2<\/sup>) has the opposite sign as x + a. The second operation in the second factor (ax + b2<\/sup>) is always positive. If the expression to be factored is a difference of cubes x3<\/sup> \u2013 a3<\/sup>, the sequence also follows SOAP. The first factor is (x – a), the same sign. The first operation (x2<\/sup> + ax) in the second factor (x2<\/sup> +ax + b2<\/sup>) has the opposite sign as (x – a) and the second operation (ax + b2<\/sup>) is always positive. The proof of factoring either the sum of cubes or the difference of cubes will be explored in more advanced college mathematics classes.<\/p>\n Figure 4: A mnemonic to remember the order of signs. (It even floats!)<\/p>\n Interested in algebra tutoring services<\/a>? Learn more about how we are assisting thousands of students each academic year.<\/p>\n SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Alabaster, AL: visit Tutoring in Alabaster, AL<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Overview Sums or differences of cubes can be factored similarly to other quadratic equations. They follow a pattern that is a little more complex than factoring quadratic equations. Sum of Cubes A sum of cubes is an expression such as x3 + a3, where both members of the expression are perfect cubes. Suppose the expression […]<\/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":[2751,2750],"class_list":["post-7209","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-difference-of-cubes","tag-sum-of-cubes"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7209","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=7209"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/7209\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=7209"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=7209"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=7209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"monohybridcross\"](\"https:\/\/schooltutoring.com\/help\/wp-content\/uploads\/sites\/2\/2014\/06\/sum-of-cubes.png\")
<\/a><\/p>\nDifference of Cubes<\/h3>\n
<\/a><\/p>\nFactoring the Sum of Cubes<\/h3>\n
Factoring the Difference of Cubes<\/h3>\n
<\/a><\/p>\nSOAP<\/h3>\n
<\/a><\/p>\n