Question

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

Answer 1

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

`x^3-24x^2+192x-512 = (x-8)^3`

Explanation:

In the general case `(a+b)^3 = a^3+3a^2b+3ab^2+b^3`

Notice that `x^3` is a perfect cube and `-512 = (-8)^3` is also a perfect cube.

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

`(x-8)^3 = x^3 + 3x^2(-8) + 3x(-8)^2 + (-8)^3`

`=x^3-24x^2+192x-512`