# How do you factor 27x ^3 + 216?

Dec 5, 2015

$27 {x}^{3} + 216 = 27 \left(x + 2\right) \left({x}^{2} - 2 x + 4\right)$

#### Explanation:

Both terms in the original expression have a factor of $27$ which can be extracted as a factor of the expression.

The remaining factor $\left({x}^{3} + 8\right) = \left({x}^{3} + {2}^{3}\right)$ is of the form
$\textcolor{w h i t e}{\text{XXX}} \left({x}^{3} + {a}^{3}\right)$
which we know has a factors $\left(x + a\right)$ and $\left({x}^{2} - a x + {a}^{2}\right)$
or in this case
$\textcolor{w h i t e}{\text{XXX}} \left({x}^{3} + 8\right) = \left(x + 2\right) \left({x}^{2} - 2 x + 4\right)$

By checking the discriminant of $\left({x}^{2} - 2 x + 4\right)$
$\textcolor{w h i t e}{\text{XXX}} {\left(- 2\right)}^{2} + 4 \left(1\right) \left(4\right) < 0 \Rightarrow$ no further Real roots