# How do you factor r^3-5r^2-100r+500?

Feb 9, 2017

The answer is $= \left(r - 10\right) \left(r - 5\right) \left(r + 10\right)$

#### Explanation:

Let $f \left(r\right) = {r}^{3} - 5 {r}^{2} - 100 r + 500$

$f \left(10\right) = 100 - 500 - 100 + 500 = 0$

So, $\left(x - 10\right)$ is a factor of $f \left(r\right)$

To find the other factors, we do a long division

$\textcolor{w h i t e}{a a a a}$${r}^{3} - 5 {r}^{2} - 100 r + 500$$\textcolor{w h i t e}{a a a a}$$|$$r - 10$

$\textcolor{w h i t e}{a a a a}$${r}^{3} - 10 {r}^{2}$$\textcolor{w h i t e}{a a a a a a a a a a a a a a a}$$|$${r}^{2} + 5 r - 50$

$\textcolor{w h i t e}{a a a a a}$$0 + 5 {r}^{2} - 100 r$

$\textcolor{w h i t e}{a a a a a a a}$$+ 5 {r}^{2} - 50 r$

$\textcolor{w h i t e}{a a a a a a a a a}$$+ 0 - 50 r + 500$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- 50 r + 500$

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a}$$- 0 + 0$

Therefore,

$f \left(r\right) = \left(r - 10\right) \left({r}^{2} + 5 r - 50\right)$

$= \left(r - 10\right) \left(r - 5\right) \left(r + 10\right)$