Chapter 4.4, Problem 16E

To determine

To factor: the polynomial completely.

Answer to Problem 16E

The factorized form is $c^3-27=\left(c-3\right)\left(c^2+3c+9\right)$ .

Explanation of Solution

Given information:

A polynomial is given as

  $c^3-27$

Calculation:

Consider the given polynomial.

  $c^3-27$

Now, factorize $c^3-27$ using the polynomial identity $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$ as shown:

  $\begin{array}{c}{c}^3-27={\left(c\right)}^3-{\left(3\right)}^3\\ =\left(c-3\right)\left\{{\left(c\right)}^2+c\cdot 3+{\left(3\right)}^2\right\}\\ =\left(c-3\right)\left(c^2+3c+9\right)\end{array}$

Therefore, $c^3-27=\left(c-3\right)\left(c^2+3c+9\right)$