Question

How do you factor the trinomial `3w^2 - 6w -24`?

Answer 1

`(3w+6)(w-4)`

Explanation:

The first term coefficient of `3` has only one option for factors namely `3 and 1`.
The last term coefficient `24` has several options but we try them all till you get the one that works out, in this case `6 and 4`.

Then you multiply across ie `1xx6=6 and 3xx4=12`.

Now from `6 and 12` you can obtain the middle term coefficient `-6` by taking `6-12`.

Then the factors will be `3w+6 and w-4`.

Answer 2

Separate out the common scalar factor `3` then find a pair of factors of `8` which differ by `2` to derive the factoring:

`3w^2-6w-24 = 3(w-4)(w+2)`

Explanation:

`3w-6w-24 = 3(w^2-2w-8) = 3(w-4)(w+2)`

In general we find: `(w+a)(w+b) = w^2+(a+b)w+ab`.

So if `w^2-2w-8` has factors with integer coefficients, `a` and `b` will be a pair of factors of `-8`, that is a pair of factors of `8` assigned different signs with `a+b = -2` and `ab = -8`.

It should not take long to spot that `a=-4`, `b=2` works.