Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 3 - Exponents, Polynomials and Functions - 3.5 Special Factoring Techniques - 3.5 Exercises: 45

Answer

$7(g^2+9h^2)(g+3h)(g-3h)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To factor the given expression, $ 7g^4-567h^4 ,$ factor first the $GCF.$ Then use the factoring of the sum or difference of $2$ squares. $\bf{\text{Solution Details:}}$ The $GCF$ of the terms is $GCF= 7 $ since it is the highest number that can evenly divide (no remainder) all the given terms. Factoring the $GCF,$ the expression above is equivalent to \begin{array}{l}\require{cancel} 7(g^4-81h^4) .\end{array} The expressions $ g^4 $ and $ 81h^4 $ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $ g^4-81h^4 ,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to \begin{array}{l}\require{cancel} 7[(g^2)^2-(9h^2)^2] \\\\= 7(g^2+9h^2)(g^2-9h^2) .\end{array} The expressions $ g^2 $ and $ 9h^2 $ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $ g^2-9h^2 ,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to \begin{array}{l}\require{cancel} 7(g^2+9h^2)[(g)^2-(3h)^2] \\\\= 7(g^2+9h^2)(g+3h)(g-3h) .\end{array}
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