Chapter 4.3, Problem 39WE

To determine

To multiply the polynomials.

Answer to Problem 39WE

The product of the expression is $a^3-3a^{2}b+3a{b}^2-b^3$

Explanation of Solution

Given information:

The given expression is ${\left(a-b\right)}^3$

Calculation:

Arrange the work horizontally.

  $\begin{array}{l}{\left(a-b\right)}^3=\left(a-b\right)\left(a-b\right)\left(a-b\right)\\ =\left(a-b\right)\left\{a\left(a-b\right)-b\left(a-b\right)\right\}\\ =\left(a-b\right)\left(a^2-ab-ab+b^2\right)\\ \text{ =}\left(a-b\right)\left(a^2-2ab+b^2\right)\\ =a\left(a^2-2ab+b^2\right)-b\left(a^2-2ab+b^2\right)\\ =a^3-2a^{2}b+a{b}^2-a^{2}b+2a{b}^2-b^3\\ =a^3-3a^{2}b+3a{b}^2-b^3\end{array}$

Therefore,

The product of the expression is $a^3-3a^{2}b+3a{b}^2-b^3$