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

Jun 7, 2018

$\left(x + 11\right) \cdot \left(3 x - 4\right)$
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: $y = a {x}^{2} + b x + c$
Start by finding 2 numbers ${x}_{1}$ and ${x}_{2}$ where:
${x}_{1} \cdot {x}_{2} = a \cdot c$
and
${x}_{1} + {x}_{2} = b$

In this case $a \cdot c = 3 \cdot \left(- 44\right) = - 132$

It usually helps to think of the prime factors of the number.
In this case, they are: $132 = 2 \cdot 2 \cdot 3 \cdot 11$
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 $- 4$
$33 \cdot \left(- 4\right) = - 132$
$33 + \left(- 4\right) = 29$

Now write down the current answers as if they were the factors:
$\left(x + 33\right) \cdot \left(x - 4\right)$

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, $3$.

$\left(x + \frac{33}{3}\right) \cdot \left(x - \frac{4}{3}\right)$
If they can be divided evenly, leave it at that. If not, move the denominator as a factor next to the x, like this:

$\left(x + 11\right) \cdot \left(3 x - 4\right)$

And those are the factors!