Chapter 4.3, Problem 46WE

To determine

To multiply the polynomials.

Answer to Problem 46WE

The product of the expression is $x^4-2x^2{y}^2+y^4$

Explanation of Solution

Given information:

The given expression is ${\left(x+y\right)}^2{\left(x-y\right)}^2$

Calculation:

Arrange the work horizontally.

  $\begin{array}{l}{\left(x+y\right)}^2{\left(x-y\right)}^2=\left\{x\left(x+y\right)+y\left(x+y\right)\right\}\left\{x\left(x-y\right)-y\left(x-y\right)\right\}\\ =\left(x^2+xy+xy+y^2\right)\left(x^2-xy-xy+y^2\right)\\ =\left(x^2+2xy+y^2\right)\left(x^2-2xy+y^2\right)\\ =x^2\left(x^2-2xy+y^2\right)+2xy\left(x^2-2xy+y^2\right)+y^2\left(x^2-2xy+y^2\right)\\ =x^4-2x^{3}y+x^2{y}^2+2x^{3}y-4x^2{y}^2+2x{y}^3+x^2{y}^2-2x{y}^3+y^4\\ =x^4-2x^2{y}^2+y^4\end{array}$

Therefore,

The product of the expression is $x^4-2x^2{y}^2+y^4$