# How do you factor  3a^2+8ab-3b^2?

Apr 21, 2016

Use an AC method to find:

$3 {a}^{2} + 8 a b - 3 {b}^{2} = \left(3 a - b\right) \left(a + 3 b\right)$

#### Explanation:

Note that this is a homogeneous polynomial: All of the terms are of degree $2$. So we can factor using a similar technique to that which we would use for a quadratic in one variable.

Use an AC Method:

Find a pair of factors of $A C = 3 \cdot 3 = 9$ which differ by $8$.

The pair $9 , 1$ works.

Use that pair to split the middle term and factor by grouping:

$3 {a}^{2} + 8 a b - 3 {b}^{2}$

$= 3 {a}^{2} + 9 a b - a b - 3 {b}^{2}$

$= \left(3 {a}^{2} + 9 a b\right) - \left(a b + 3 {b}^{2}\right)$

$= 3 a \left(a + 3 b\right) - b \left(a + 3 b\right)$

$= \left(3 a - b\right) \left(a + 3 b\right)$