Chapter 4.5, Problem 48WE

To determine

To factor thepolynomial.

Answer to Problem 48WE

The value of polynomial is $8ab\left(a^2+b^2\right)$ .

Explanation of Solution

Given information:

  ${\left(a+b\right)}^4-{\left(a-b\right)}^4$

Calculation:

The polynomial can be obtained as

Let $\left(a+b\right)=x\text{ and }\left(a-b\right)=y$

  $\begin{array}{l}{\left(a+b\right)}^4-{\left(a-b\right)}^4\\ =\left[{\left({\left(x\right)}^2\right)}^2-{\left({\left(y\right)}^2\right)}^2\right]\left[\text{use of power property}\right]\\ =\left[\left(x^2+y^2\right)\left(x^2-y^2\right)\right]\left[\text{difference of squares}\right]\\ =\left[\left(x^2+y^2\right)\left(x+y\right)\left(x-y\right)\right]\left[\text{difference of squares }\left(x^2-y^2\right)\right]\\ =\left[\left(x^2+y^2\right)\left(x+y\right)\left(x-y\right)\right]\text{ equation 1}\end{array}$

First find the values of $\left(x+y\right)\left(x-y\right)$

Put the values of $x=\left(a+b\right)\text{ and y=}\left(a-b\right)$

  $\begin{array}{l}\left(x+y\right)\left(x-y\right)\\ =\left[\left(a+b\right)+\left(a-b\right)\right]\left[\left(a+b\right)-\left(a-b\right)\right]\\ =\left(a+b+a-b\right)\left(a+b-a+b\right)\\ =\left(2a\right)\left(2b\right)\\ =4ab\end{array}$

Again, find the values of $\left(x^2+y^2\right)$

   put the values of $x=\left(a+b\right)\text{ and y=}\left(a-b\right)$

  $\begin{array}{l}\left(x^2+y^2\right)\\ =\left[{\left(a+b\right)}^2+{\left(a-b\right)}^2\right]\\ =\left[\left(a^2+2ab+b^2\right)+\left(a^2-2ab+b^2\right)\right]\left[\text{perfect square trinomial}\right]\\ =\left(a^2+2ab+b^2+a^2-2ab+b^2\right)\left[\text{simplify}\right]\\ =\left(2a^2+2b^2\right)\\ =2\left(a^2+b^2\right)\left[\text{take out the common number 2}\right]\end{array}$

Put the values of first and second part in Equation 1

  $\begin{array}{l}2\left(a^2+b^2\right)\left(4ab\right)\\ =8ab\left(a^2+b^2\right)\end{array}$

Hence,

The value of polynomial is $8ab\left(a^2+b^2\right)$ .