## Intermediate Algebra: Connecting Concepts through Application

$7g^3h^2 \left( 3g^4-g^2h^2+20h^4 \right)$
$\bf{\text{Solution Outline:}}$ Get the $GCF$ of the given expression, $21g^7h^2-7g^5h^4+140g^3h^6 .$ Divide the given expression and the $GCF.$ Express the answer as the product of the $GCF$ and the resulting quotient. $\bf{\text{Solution Details:}}$ The $GCF$ of the constants of the terms $\{ 21,-7,140 \}$ is $7$ since it is the highest number that can evenly divide (no remainder) all the given constants. The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{ g^7h^2,g^5h^4,g^3h^6 \}$ is $g^3h^2 .$ Hence, the entire expression has $GCF= 7g^3h^2 .$ Factoring the $GCF= 7g^3h^2 ,$ the expression above is equivalent to \begin{array}{l}\require{cancel} 7g^3h^2 \left( \dfrac{21g^7h^2}{7g^3h^2}-\dfrac{7g^5h^4}{7g^3h^2}+\dfrac{140g^3h^6}{7g^3h^2} \right) \\\\= 7g^3h^2 \left( 3g^4-g^2h^2+20h^4 \right) .\end{array}