# 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)

-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, 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

-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. x2 −x −20 = (x + 4) * (x + -5)

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