How do you factor trinomials?

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How do you factor trinomials?

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Answer 1 · 2015-01-04T21:21:35

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 $x^{2} + 15 x + 36$ $x^2+15x+36$
Then you try to write $36$ $36$ as the product of two numbers, and $15$ $15$ 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 $36 = 1 \cdot 36 = 2 \cdot 18 = 3 \cdot 12 = 4 \cdot 9 = 6 \cdot 6$ $36=1*36=2*18=3*12=4*9=6*6$
Sums of these are $37, 20, 15, 13, 12$ $37,20,15,13,12$ respectively
Differences are $35, 16, 9, 5, 0$ $35,16,9,5,0$ respectively
$15 = + 3 + 12$ $15=+3+12$ will do. So the factoring becomes:
$(x + 3) (x + 12)$ $(x+3)(x+12)$
Check your answer! $= x^{2} + 12 x + 3 x + 36$ $=x^2+12x+3x+36$

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

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

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

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How do you factor trinomials?

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