# How do you factor trinomials?

Jan 4, 2015

Extension to factoring, when the trinomials do not factor into a square (it also works with squares).

Sum-product-method
Say you have an expression like ${x}^{2} + 15 x + 36$
Then you try to write $36$ as the product of two numbers, and $15$ as the sum (or difference) of the same two numbers. In this case (with both being positive) it's not so hard. You take the sum.

You can write $36 = 1 \cdot 36 = 2 \cdot 18 = 3 \cdot 12 = 4 \cdot 9 = 6 \cdot 6$
Sums of these are $37 , 20 , 15 , 13 , 12$ respectively
Differences are $35 , 16 , 9 , 5 , 0$ respectively
$15 = + 3 + 12$ will do. So the factoring becomes:
$\left(x + 3\right) \left(x + 12\right)$
Check your answer! $= {x}^{2} + 12 x + 3 x + 36$

It's a bit harder when one or two of the numbers are negative, let's take ${x}^{2} - 15 x + 36$
Same as the first, only now both factors are negative
$\left(x - 3\right) \left(x - 12\right) = {x}^{2} - 12 x - 3 x + 36 =$ the original

Extra
If the last number ($36$) is negative, you will have to work with the difference of the factors. Check the next one yourself:
x^2+5x-36=(x+9)(x-4)=?

And now try: x^2-5x-36=?