# How do you factor (2x - 3)^3 - 27?

May 9, 2015

Factoring ${\left(2 x - 3\right)}^{3} - 27$

Consider a related problem:
Find the solutions for ${\left(2 x - 3\right)}^{3} - 27 = 0$

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

${\left(2 x - 3\right)}^{3} = {3}^{3}$

$2 x - 3 = 3$

$x = 3$

So $\left(x - 3\right)$ is a factor of ${\left(2 x - 3\right)}^{3} - 27$

Using synthetic division we can determine that
${\left(2 x - 3\right)}^{3} - 27 = \left(x - 3\right) \left(8 {x}^{2} - 12 x + 18\right)$

We can extract a common factor of $\left(2\right)$ from this final factor
${\left(2 x - 3\right)}^{3} - 27 = \left(x - 3\right) \left(2\right) \left(4 {x}^{2} - 6 x + 9\right)$

Examining the discriminant of $\left(4 {x}^{2} - 6 x + 9\right)$ reveals that there are no further factors.