Factoring trinomials
A trinomial is a polynomial made up of three terms. Factoring trinomials is the inverse of multiplying two binomials
Instead of multiplying two binomials to get a trinomial, you will write the trinomial as a product of two binomials
The general form of a trinomial is ax
^{2} + bx + c
Your goal in factoring trinomials is to make ax
^{2} + bx + c equal to (? + ?) * (? + ?)
When a = 1, the trinomial becomes x
^{2} + bx + c and it is easier to factor. Therefore, I will start by showing you
how to factor when a = 1.
Example #1:
Factor x
^{2} + 5x + 6
x
^{2} + 5x + 6 will look like (x + ?) * (x + ?)
We are 100 % sure that the first term for each binomial must be x because x * x = x
^{2}
Now, how do we get the second term for each binomial?
We also know for sure that ? * ? or the product of the second term for each binomial is equal to 6
Finally, we know that x * ? and ? * x must give the second term, which is 5x when added
Thus,when factoring trinomials, the trick is to look for factors of 6(last term), that will add up to 5(coefficient of second term)
-6 is equal to:
6 × 1
-6 × - 1
2 × 3
-2 × -3
The only pair of factors that will add up to 5 is 2 and 3 because 2 + 3 = 5
Just replace the two question mark by 2 and 3 and you are done
Therefore, x
^{2} + 5x + 6 = (x + 3) * (x + 2)
Notice that (x + 3) * (x + 2) = also equal to (x + 2) * (x + 3) since multiplication is commutative
The final step is to check your answer by multiplying the two binomials
x * x = x
^{2}
x * 2 = 2x
3 * x = 3x
3 * 2 = 6
Since 2x + 3x = 5x, putting it all together, we get:
x
^{2} + 5x + 6
Example #2:
Factor x
^{2} −5x + 6
It is almost the same equation as before with the exception that the coefficient of the second term is -5 instead
of 5
Follow all steps outlined above. The only difference is that you will be looking for factors of 6 that will add up
to -5 instead of 5.
-3 and -2 will do the job
So, x
^{2} −5x + 6 = (x + -3) * (x + -2)
Final example
Factor x
^{2} −x −20
First, notice that x
^{2} −x −20 = x
^{2} −1x −20 because 1*x = x
x
^{2} −x −20 = (x + ?) * (x + ?)
Find factors of -20 that will equal to -1
-20 is equal to
-20 * 1
20 * -1
10 * -2
-10 * 2
4 * -5
-4 * 5
Since 4 + -5 = -1, we have found what we need.
x
^{2} −x −20 = (x + 4) * (x + -5)