Question

How do you factor `6k^2 + 5kp - 6p^2`?

Answer 1

`6k^2+5kp-6p^2 = (3k-2p)(2k+3p)`

Explanation:

Given:

`6k^2+5kp-6p^2`

Note that this is a homogeneous polynomial of degree `2` - that is all of the terms are of degree `2`.

To factor it, we can treat it similarly to a quadratic in one variable and use an AC method:

Look for a pair of factors of `AC=6*6=36` which differ by `B=5`.

The pair `9, 4` works.

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

`6k^2+5kp-6p^2 = 6k^2+9kp-4kp-6p^2`

`color(white)(6k^2+5kp-6p^2) = (6k^2+9kp)-(4kp+6p^2)`

`color(white)(6k^2+5kp-6p^2) = 3k(2k+3p)-2p(2k+3p)`

`color(white)(6k^2+5kp-6p^2) = (3k-2p)(2k+3p)`