# How do you factor 5r ^ { 3} s + 30r ^ { 2} - 5r ^ { 2} s - 30r ^ { 3}?

Jun 26, 2018

=$5 {r}^{2} \left(r - 1\right) \left(s - 6\right)$

#### Explanation:

First we notice that $5 {r}^{2}$ is an element in each term, so that
$5 {r}^{2} s + 30 {r}^{2} - 5 {r}^{s} - 30 {r}^{3}$
=$5 {r}^{2} \left(r s + 6 - s - 6 r\right)$
We rearrange:
=5r^2((rs-6r) -(s-6)
=5r^2(r(s-6) -(s-6)
=$5 {r}^{2} \left(r - 1\right) \left(s - 6\right)$

Jun 26, 2018

$5 {r}^{3} s + 30 {r}^{2} - 5 {r}^{2} s - 30 {r}^{3} = 5 {r}^{2} \left(s - 6\right) \left(r - 1\right)$

#### Explanation:

$5 {r}^{3} s + 30 {r}^{2} - 5 {r}^{2} s - 30 {r}^{3} = 5 {r}^{3} s - 5 {r}^{2} s - 30 {r}^{3} + 30 {r}^{2}$
Factoring by grouping, we get
$5 {r}^{3} s - 5 {r}^{2} s - 30 {r}^{3} + 30 {r}^{2} = 5 {r}^{2} s \left(r - 1\right) - 30 {r}^{2} \left(r - 1\right)$
$5 {r}^{2} s \left(r - 1\right) - 30 {r}^{2} \left(r - 1\right) = \left(5 {r}^{2} s - 30 {r}^{2}\right) \left(r - 1\right)$
Factoring out $5 {r}^{2}$ we get
$\left(5 {r}^{2} s - 30 {r}^{2}\right) \left(r - 1\right) = 5 {r}^{2} \left(s - 6\right) \left(r - 1\right)$
$\therefore 5 {r}^{3} s + 30 {r}^{2} - 5 {r}^{2} s - 30 {r}^{3} = 5 {r}^{2} \left(s - 6\right) \left(r - 1\right)$