Question

How do you factor `y= 3x^2+29x-44` ?

Answer 1

`(x+11)*(3x-4)`
The explanation below shows one method on how to factor polynomials when the leading coefficient is not equal to 1.

Explanation:

Here's one way to factor a polynomial when the leading coefficient not equal to 1:

For: `y=ax^2+bx+c`
Start by finding 2 numbers `x_1` and `x_2` where:
`x_1 * x_2 = a*c`
and
`x_1 + x_2 = b`

In this case `a*c = 3*(-44) = -132`

It usually helps to think of the prime factors of the number.
In this case, they are: `132 = 2*2*3*11`
You can try a few combinations, keeping in mind that, in this case, the 2 numbers must add to `29`.
The winning combination turns out to be `33` and `-4`
`33*(-4) = -132`
`33+(-4) = 29`

Now write down the current answers as if they were the factors:
`(x+33)*(x-4)`

However, that's not quite the answer yet. If we left it like this, the leading coefficient would be 1 and the last coefficient would be too big. So we need to adjust for that.

Divide the 2 answers by the leading coefficient, in this case, `3`.

`(x+33/3)*(x-4/3)`
If they can be divided evenly, leave it at that. If not, move the denominator as a factor next to the x, like this:

`(x+11)*(3x-4)`

And those are the factors!