# How do you factor 25 g^3 v^3 + 15 g^2 e^3 v^3 + 5 g b d + 3 e^3 b d?

Jun 22, 2016

$25 {g}^{3} {v}^{3} + 15 {g}^{2} {e}^{3} {v}^{3} + 5 g b d + 3 {e}^{3} b d = \textcolor{b l u e}{\left(5 {g}^{2} {v}^{3} + b d\right) \left(5 g + 3 {e}^{3}\right)}$

#### Explanation:

Note 1:
$\textcolor{w h i t e}{\text{XXX}}$The first two terms have a common factor of
$\textcolor{w h i t e}{\text{XXXXXX}} 5 {g}^{2} {v}^{3}$
$\textcolor{w h i t e}{\text{XXX}} 25 {g}^{3} {v}^{3} + 15 {g}^{2} {e}^{3} {v}^{3} = \left(\textcolor{red}{5 {g}^{2} {v}^{3}}\right) \left(5 g + 3 {e}^{3}\right)$

Note 2:
$\textcolor{w h i t e}{\text{XXX}}$The last two terms have a common factor of
$\textcolor{w h i t e}{\text{XXXXXX}} b d$
$\textcolor{w h i t e}{\text{XXX}} 5 g b d + 3 {e}^{3} b d = \left(\textcolor{g r e e n}{b d}\right) \left(5 g + 3 {e}^{3}\right)$

Therefore:
$\textcolor{w h i t e}{\text{XXX}} 25 {g}^{3} {v}^{3} + 15 {g}^{2} {e}^{3} {v}^{3} + 5 g b d + 3 {e}^{3} b d$
$\textcolor{w h i t e}{\text{XXXXXX}} = \left(\textcolor{red}{5 {g}^{2} {v}^{3}}\right) \left(5 g + 3 {e}^{3}\right) + \left(\textcolor{g r e e n}{b d}\right) \left(5 g + 3 {e}^{3}\right)$

Note 3:
$\textcolor{w h i t e}{\text{XXX}}$The second factor for each term is identical

$\textcolor{w h i t e}{\text{XXX}} 25 {g}^{3} {v}^{3} + 15 {g}^{2} {e}^{3} {v}^{3} + 5 g b d + 3 {e}^{3} b d$
$\textcolor{w h i t e}{\text{XXXXXX}} = \left(\textcolor{red}{5 {g}^{2} {v}^{3}} + \textcolor{g r e e n}{b d}\right) \left(5 g + 3 {e}^{3}\right)$