# How do you factor x^3 - 24x^2 + 192x - 512?

Oct 4, 2015

Notice that the leading and trailing terms are both perfect cubes, so try and find that:

${x}^{3} - 24 {x}^{2} + 192 x - 512 = {\left(x - 8\right)}^{3}$

#### Explanation:

In the general case ${\left(a + b\right)}^{3} = {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + {b}^{3}$

Notice that ${x}^{3}$ is a perfect cube and $- 512 = {\left(- 8\right)}^{3}$ is also a perfect cube.

So try $a = x$ and $b = - 8$ to find:

${\left(x - 8\right)}^{3} = {x}^{3} + 3 {x}^{2} \left(- 8\right) + 3 x {\left(- 8\right)}^{2} + {\left(- 8\right)}^{3}$

$= {x}^{3} - 24 {x}^{2} + 192 x - 512$