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 ax² + bx + c

Your goal in factoring trinomials is to make ax² + bx + c equal to (? + ?) * (? + ?)

When a = 1, the trinomial becomes x² + bx + c and it is easier to factor. Therefore, I will start by showing you how to factor when a = 1.

Example #1:

Factor x² + 5x + 6

x² + 5x + 6 will look like (x + ?) * (x + ?)

We are 100 % sure that the first term for each binomial must be x because x * x = x²

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)

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, x² + 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 = x²

x * 2 = 2x

3 * x = 3x

3 * 2 = 6

Since 2x + 3x = 5x, putting it all together, we get:

x² + 5x + 6

Example #2:

Factor x² −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, x² −5x + 6 = (x + -3) * (x + -2)

Final example

Factor x² −x −20

First, notice that x² −x −20 = x² −1x −20 because 1*x = x

x² −x −20 = (x + ?) * (x + ?)

Find factors of -20 that will equal to -1

Since 4 + -5 = -1, we have found what we need. x² −x −20 = (x + 4) * (x + -5)

Try this factoring trinomials quiz to see how well you can factor trinomials.