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

Jun 11, 2015

Spotted $- {x}^{3} + 15 {x}^{2} - 75 x + 125 = {\left(5 - x\right)}^{3}$

Explanation:

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

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

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

${\left(5 - x\right)}^{3} = {5}^{3} - 3 \cdot {5}^{2} x + 3 \cdot 5 x - {x}^{3}$

$= 125 - 75 x + 15 {x}^{2} - {x}^{3}$