Factoring Trinomials

A trinomial is a polynomial that has three terms. A polynomial has two or more algebraic terms. The first term has an exponent to the second degree (x²), the second term is x, and the last term is a constant (just any number). Factoring trinomials is an extremely important algebra skill. Factoring polynomials can be tricky however with the detailed explanation below any problem can be solved.

Trinomial

ax² + bx + c

a = the x² term

b = the x term

c = the constant value

Example 1:

a² + 4a – 12

First identify the values for a, b, and c.

a = 1

b = 4

c = -12

Remember when no number is in front of a variable, we assume the coefficient is 1.

Step 2: Find two numbers that when added together produce b and when multiplied together produce c. In this case we should list the factors of -12 first.

Factors of -12:

Possible Combinations	Does c = -12 when multiplied?	Does b = 4 when added?	Correct Combination
4 x 3	12	7	No
4 x -3	-12	1	No
-4 x -3	12	-7	No
-4 x 3	-12	-1	No
6 x 2	12	8	No
-6 x 2	-12	-4	No
6 x -2	-12	4	Yes
-6 x -2	12	-8	No
12 x 1	12	13	No
-12 x 1	-12	-11	No
12 x -1	-12	11	No
-12 x -1	12	-13	No

The table above shows all the possible combinations for example 1. However only one combination equals both b and c (highlighted in bold). The correct combination of the two numbers should look like this:

(a – 2) (a + 6)

We can foil the problem to see if it matches the original problem.

Example 2:

Factor the following polynomial:

x² + 2x -8

First identify the values of a, b, and c.

a = 1

b = 2

c =-8

Now consider the factors of c and when added together equal b.

Possible Combinations	Does c = -8 when multiplied?	Does b = 2 when added?	Correct Combination
8 x 1	8	9	No
-8 x -1	8	-9	No
8 x -1	-8	7	No
-8 x 1	8	-7	No
2 x 4	8	6	No
-2 x 4	-8	2	Yes
2 x -4	-8	-2	No
-2 x -4	8	-8	No