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
NonExamples of Quadratic Trinomials:

3x^{3} + 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

5x^{3} + 6x^{2} + 9
this is not a quadratic trinomial because there is an exponent that is greater than 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 x^{2} is 1.
 identify a,b, and c in the trinomial ax^{2} + bx+c
 write down all factor pairs of c
 identify which factor pair from the previous step sums up to b
 Substitute factor pairs into two binomials
Example of Factoring a Trinomial
Example 1
Factor x^{2} + 5x + 4
Step 1Identify a,b, and c in the trinomial
ax^{2} + bx+c
a=
1
b=
5
c=
4
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)
identify which factor pair from the previous step sums up to c
If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial
Practice Problems
identify a,b, and c in the trinomial ax^{2} + bx+c
a =
1
b =
4
c = 3
write down all factor pairs of 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)
Substitute that factor pair into two binomials
(x +3)(x+1)
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) = x^{2} + 4x + 3
identify a,b, and c in the trinomial ax^{2} + bx+c
a=
1
b=
5
c=
6
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)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)
Substitute that factor pair into two binomials
(x +2)(x+3)
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) = x^{2} + 5x + 6
identify a,b, and c in the trinomial ax^{2} + bx+c
a = 1
b = 2
c = 3
write down all factor pairs of 3 (yes, the negative sign matters!)
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)Substitute that factor pair into two binomials
(x + 1 )(x  3)
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) = x^{2} 2x 3
identify a,b, and c in the trinomial ax^{2} + bx+c
a = 1
b = 5
c = 6
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)
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)
Substitute that factor pair into two binomials
(x 2 )(x  3)
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) = x^{2 }5x +6
identify a,b, and c in the trinomial ax^{2} + bx+c
a = 1
b = 5
c = 6
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)
identify which factor pair from the previous step sums up to 2
Substitute that factor pair into two binomials
(x +3 )(x 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)= x^{2 }2x 15

Related Links:
 Solving Quadratics
 Methods of Factoring
 How to Factor by Grouping