## Intermediate Algebra: Connecting Concepts through Application

$ab^2(a-b)(24a+35b)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $24a^3b^2+11a^2b^3-35ab^4 ,$ factor first the $GCF.$ Then find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping. $\bf{\text{Solution Details:}}$ The $GCF$ of the terms is $GCF= ab^2$ since it is the highest expression that can evenly divide (no remainder) all the given terms. Factoring the $GCF,$ the expression above is equivalent to \begin{array}{l}\require{cancel} ab^2(24a^2+11ab-35b^2) .\end{array} Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $24(-35)=-840$ and the value of $b$ is $11 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ -24,35 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} ab^2(24a^2-24ab+35ab-35b^2) .\end{array} Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to \begin{array}{l}\require{cancel} ab^2[(24a^2-24ab)+(35ab-35b^2)] .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} ab^2[24a(a-b)+35b(a-b)] .\end{array} Factoring the $GCF= (a-b)$ of the entire expression above results to \begin{array}{l}\require{cancel} ab^2[(a-b)(24a+35b)] \\\\= ab^2(a-b)(24a+35b) .\end{array}