Question

How do you factor `-x^3 + 15x^2 - 75x + 125`?

Answer 1

Spotted `-x^3+15x^2-75x+125 = (5-x)^3`

Explanation:

Noticing that the first term is `(-x)^3` and the last is `5^3`, I immediately considered the possibility that this is a perfect cube quadrinomial - in fact `(5-x)^3`.

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

If we put `a = 5` and `b = x` then

`(5-x)^3 = 5^3-3*5^2x+3*5x-x^3`

`=125-75x+15x^2-x^3`