Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter P - Section P.5 - Factoring Polynomials - Exercise Set - Page 70: 140


The required expression is $\underline{\left( x+y \right){{\left( x-y \right)}^{3}}}$.

Work Step by Step

Let us consider the given polynomial: ${{x}^{4}}-{{y}^{4}}-2{{x}^{3}}y+2x{{y}^{3}}$ …………..….. (1) Factorize it with the help of the technique, ‘‘factoring the polynomial by grouping’’: $\begin{align} & {{x}^{4}}-{{y}^{4}}-2{{x}^{3}}y+2x{{y}^{3}}=\left( {{x}^{4}}-{{y}^{4}} \right)+\left( -2{{x}^{3}}y+2x{{y}^{3}} \right) \\ & =\left( {{\left( {{x}^{2}} \right)}^{2}}-{{\left( {{y}^{2}} \right)}^{2}} \right)+\left( -2{{x}^{3}}y+2x{{y}^{3}} \right) \\ & =\left( {{x}^{2}}+{{y}^{2}} \right)\left( {{x}^{2}}-{{y}^{2}} \right)-2xy\left( {{x}^{2}}-{{y}^{2}} \right) \\ & =\left( {{x}^{2}}-{{y}^{2}} \right)\left( {{x}^{2}}+{{y}^{2}}-2xy \right) \\ & =\left( x+y \right)\left( x-y \right){{\left( x-y \right)}^{2}} \\ & =\left( x+y \right){{\left( x-y \right)}^{3}} \end{align}$
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