Chapter 4.4, Problem 13E

To determine

To factor: the polynomial completely.

Answer to Problem 13E

The factorized form is $x^3+64=\left(x+3\right)\left(x^2-4x+16\right)$ .

Explanation of Solution

Given information:

A polynomial is given as

  $x^3+64$

Calculation:

Consider the given polynomial.

  $x^3+64$

Now, factorize $x^3+64$ using the polynomial identity $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$ as shown:

  $\begin{array}{c}{x}^3+64={\left(x\right)}^3+{\left(4\right)}^3\\ =\left(x+3\right)\left\{{\left(x\right)}^2-x\cdot 4+{\left(4\right)}^2\right\}\\ =\left(x+3\right)\left(x^2-4x+16\right)\end{array}$

Therefore, $x^3+64=\left(x+3\right)\left(x^2-4x+16\right)$