Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 5 - Section 5.1 - Greatest Common Factors and Factoring by Grouping - 5.1 Exercises - Page 329: 34


$(5-x)^2 \left( 7-2x \right)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Get the $GCF$ of the given expression, $ 2(5-x)^3-3(5-x)^2 .$ 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 $\{ 2,-3 \}$ is $ 1 .$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variables $\{ (5-x)^3,(5-x)^2 \}$ is $ (5-x)^2 .$ Hence, the entire expression has $GCF= (5-x)^2 .$ Factoring the $GCF= (5-x)^2 ,$ the given expression is equivalent to \begin{array}{l}\require{cancel} (5-x)^2 \left( \dfrac{2(5-x)^3}{(5-x)^2}-\dfrac{3(5-x)^2}{(5-x)^2} \right) .\end{array} Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to \begin{array}{l}\require{cancel} (5-x)^2 \left( 2(5-x)^{3-2}-3(5-x)^{2-2} \right) \\\\= (5-x)^2 \left( 2(5-x)^{1}-3(5-x)^{0} \right) \\\\= (5-x)^2 \left( 2(5-x)-3(1) \right) \\\\= (5-x)^2 \left( 10-2x-3 \right) \\\\= (5-x)^2 \left( 7-2x \right) .\end{array}
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