# How To Factor Trinomials

Step by Step Tutorial

#### What is a Trinomial?

A trinomial is a polynomial with 3 terms.. This page will focus on quadratic trinomials. The degree of a quadratic trinomial must be '2'. In other words, there must be an exponent of '2' and that exponent must be the greatest exponent.

 $$3x^2 + 2x + 1$$ $$7 x^2 + 4x + 4$$ $$5 x^2 + 6x + 9$$
 3x3 + 2x + 1 this is not a quadratic trinomial because there is an exponent that is greater than 2 2x + 4 this is not a quadratic trinomial because there is not exponent of 2. In fact, this is not even a trinomial because there are 2 terms 5x3 + 6x2 + 9 this is not a quadratic trinomial because there is an exponent that is greater than 2
Note: For the rest of this page, 'factoring trinomials' will refer to factoring 'quadratic trinomials' . (The only difference being that a quadratic trinomial has a degree of 2.)

### Formula For Factoring Trinomials(when a =1)

It's always easier to understand a new concept by looking at a specific example so you might want to do that first. This formula works when 'a' is 1. In other words, we will use this approach whenever the coefficient in from of x2 is 1. (If you need help factoring trinomials when $$a \ne 1$$, then go here.

1. identify a,b, and c in the trinomial ax2 + bx+c
2. write down all factor pairs of c
3. identify which factor pair from the previous step sums up to b
4. Substitute factor pairs into two binomials

### Example of Factoring a Trinomial

##### Example 1

Factor x2 + 5x + 4

Step 1

Identify a,b, and c in the trinomial
ax2 + bx+c

a= 1
b= 5
c= 4

Step 2

Write down all factor pairs of 4
(Note: since 5 is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive. As the chart on the right shows you -2*-2 is positive 4...so we do have to consider these two negative factors. This is probably easier to understand if you watch our video lesson factoring trinomials)

Step 3

identify which factor pair from the previous step sums up to c

Step 4
Substitute that factor pair into two binomials
(x +4)(x+1)
Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

(x +4)(x+1) = x2 + 5x + 4

### Practice Problems

Step 1

identify a,b, and c in the trinomial ax2 + bx+c

a = 1
b = 4
c = 3

Step 2

write down all factor pairs of 3

Step 3

identify which factor pair from the previous step sums up to 4 (Note: since 4 is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive)

Step 4

Substitute that factor pair into two binomials

(x +3)(x+1)

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

(x +3)(x+1) = x2 + 4x + 3

Step 1

identify a,b, and c in the trinomial ax2 + bx+c

a= 1
b= 5
c= 6

Step 2

write down all factor pairs of 6

(Note: since 6 is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive)
Step 3

identify which factor pair from the previous step sums up to 5 (Note: since 5 is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive)

Step 4

Substitute that factor pair into two binomials

(x +2)(x+3)

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

(x +2)(x+3) = x2 + 5x + 6

Step 1

identify a,b, and c in the trinomial ax2 + bx+c

a = 1
b = -2
c = -3

Step 2

write down all factor pairs of -3 (yes, the negative sign matters!)

Step 3

identify which factor pair from the previous step sums up to -2

(Note: since c is negative we only need to think about pairs that have 1 negative factor and 1 positive factor. Remember a negative times a positive is a negative)
Step 4

Substitute that factor pair into two binomials

(x + 1 )(x - 3)

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

(x +2)(x- 3) = x2- 2x -3

Step 1

identify a,b, and c in the trinomial ax2 + bx+c

a = 1
b = -5
c = 6

Step 2

write down the factor pairs of 6
(Note: since 6 is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive)

Step 3

identify which factor pair from the previous step sums up to -5
(Note: since c is negative we only need to think about pairs that have 1 negative factor and 1 positive factor. Remember a negative times a positive is a negative)

Step 4

Substitute that factor pair into two binomials

(x -2 )(x - 3)

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

(x -2)(x- 3) = x2 -5x +6

Step 1

identify a,b, and c in the trinomial ax2 + bx+c

a = 1
b = -2
c = -15

Step 2

write down the factor pairs of -15
(Note: since c is negative we only need to think about pairs that have 1 negative factor and 1 positive factor. Remember a negative times a positive is a negative)

Step 3

identify which factor pair from the previous step sums up to -2

Step 4

Substitute that factor pair into two binomials

(x +3 )(x- 5)

Step 5

If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial

(x +3 )(x- 5)= x2 -2x -15