How To Factor Trinomials

Step by Step Tutorial

What is a Trinomial?

Answer:

First, some definitions and terminology

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.

Examples of Quadratic Trinomials:
  • 3x² + 2x + 1
  • 7x² + 4x + 4
  • 5x² + 6x + 9

Non-Examples of Quadratic Trinomials:
  • 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.)

Video Tutorial

on Factoring Trinomials

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.

  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)

factor table picture
Step 3

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

factor table picture
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

Problem 1

Factor the following trinomial: x2+4x +3

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

factor table picture
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)

factor table picture
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

Problem 2

Factor the trinomial below x2+5x +6

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)
factor table picture
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)

factor table picture
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

Problem 3

Factor the trinomial below x2 - 2x -3

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!)

factor table picture
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)
factor table picture
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

Problem 4

Factor the trinomial below x2 -5x +6

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)

factor table picture
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)

factor table picture
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

Problem 5

Factor the trinomial below x2 -2x -15

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 -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)

factor table picture
Step 3

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

factor table picture
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