Chapter 8.4, Problem 53PPS

To determine

To find: The product ${\left(n-p\right)}^2\left(n+p\right)$

Answer to Problem 53PPS

  $n^3-n{p}^2-n^{2}p+p^3$

Explanation of Solution

Given:

  ${\left(n-p\right)}^2\left(n+p\right)$

Calculation:

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

Rewrite this expression as

  $\left(n-p\right)\left(n-p\right)\left(n+p\right)$

Use the formula $\left(a+b\right)\left(a-b\right)=a^2-b^2$ and then multiply using the distributive property

  $\begin{array}{l}\left(n-p\right)\left(n-p\right)\left(n+p\right)\\ =\left(n-p\right)\left(n^2-p^2\right)\\ =n^3-n{p}^2-n^{2}p+p^3\end{array}$