# How do you factor  6uv - 3v^3 + 4u - 2v^2?

May 3, 2015

This method will not work in all cases but it is one way to attempt to find a solution.

Look for two pair of terms with the same ratio between the coefficients of the elements of the pairs.
In this case there are two possibilities:
$6 : - 3 \equiv 4 : - 2$
and
$\textcolor{red}{6 : 4} \equiv \textcolor{b l u e}{- 3 : - 2}$

Examine each for possible extraction of common factors

The first possibilities doesn't have any useful factors (other than 2) in the terms $4 u$ and $- 2 {v}^{2}$

So let's try the other pairing
$6 u v - 3 {v}^{3} + 4 u - 2 {v}^{2}$

$= \textcolor{red}{6 u v + 4 u} - \textcolor{b l u e}{3 {v}^{3} + 2 {v}^{2}}$

$= \textcolor{red}{2 u \left(3 v + 2\right)} - \textcolor{b l u e}{{v}^{2} \left(3 v + 2\right)}$

$= \left(2 u - {v}^{2}\right) \left(3 v + 2\right)$