How do you factor $6 u v - 3 v^{3} + 4 u - 2 v^{2}$ $6uv - 3v^3 + 4u - 2v^2$ ?

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How do you factor $6 u v - 3 v^{3} + 4 u - 2 v^{2}$ $6uv - 3v^3 + 4u - 2v^2$ ?

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Answer 1 · 2015-05-03T18:06:04

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:
$6 : - 3 \equiv 4 : - 2$ $6:-3 -= 4:-2$
and
$6 : 4 \equiv - 3 : - 2$ $color(red)(6:4) -= color(blue)(-3:-2)$

Examine each for possible extraction of common factors

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

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

$= 6 u v + 4 u - 3 v^{3} + 2 v^{2}$ $=color(red)(6uv+4u) - color(blue)(3v^3+2v^2)$

$= 2 u (3 v + 2) - v^{2} (3 v + 2)$ $= color(red)(2u(3v+2)) - color(blue)(v^2(3v+2))$

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

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How do you factor $6 u v - 3 v^{3} + 4 u - 2 v^{2}$ $6uv - 3v^3 + 4u - 2v^2$ ?

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