Factoring Trinomials

By Carolyn Rutman

Trinomials are 3-termed expressions.

We are going to look at trinomials of the form ax2 + bx + c, where a = 1.

Examples: x2 + 5x + 4

x2 – 7x + 10

x2 + 6x – 16

Let’s review how to multiply binomials.

(x + 4)(x + 2) =

x2 + 2x

+ 4x+ 8 =

x2 + 6x + 8Notice that the first term of the trinomial comes from the first terms in the parentheses.

The last term comes from the two last numbers in the parentheses.

We are going to use that idea to help us factor trinomials.

Let’s factor this trinomial, step by step. x2 + 11x

+ 18First, set up two parentheses and put x as the first term in each.

Why? Because x times x gives us the

x2. ( x ) (x )

x2 + 11x + 18(x )(x )

Now determine what kind of signs you will need.

Since you want the last two numbers in the parentheses to multiply to give + 18, you will need two numbers with the same sign.

Remember: + times + = + - times - = +

x2 + 11x + 18(x )(x )

We want two signs that are the same to get a +18. Should it be two +’s or two –’s?

The middle term, 11x, is positive, so we want two + signs. **Remember, two positives multipy to give a positive and add to give a positive.

+ +

x2 + 11x + 18(x + )(x + )

Now that we have the x’s and the signs, we need to find two numbers which will multiply to give 18.

Because the signs are the same, these numbers must also add to give 11.

( Notice the + sign at the back is + . That will help you remember the numbers must add, not subtract, to give 11. )

+

Your choices are 1, 18 or 2, 9 or 3, 6 .

2 and 9 will do the trick!

So we factor the trinomial x2 + 11x + 18 (x + 9) (x + 2)

Check this by multiplying the binomials to see if the product is what you started

with.

Here is another one to factor: x2 – 9x + 14(x )(x ) Set up ( ) and put in the x’s.

(x - )(x - ) Determine the signs. (You have a product of +14, so you need like signs. Since the middle term is – 5 , you need two –

signs.

(x – 7)(x – 2) Find 2 numbers which multiply to give 14 and add to give 9. 2 and 7 work.

Check your answer by multiplying the binomials.

Look at a trinomial with the last sign – .

x2 + 6x – 27(x ) (x ) Set up the ( ) and x’s.

(x + )(x - ) Determine the signs. Since the product is – 27, we need to use

two different signs. Put + and - in the parentheses.

(x + 9)(x – 3) Find 2 numbers which multiply to give 27 and subtract to give 6.

9 and 3 work (We will need +9 and -3 to get a +6.)

Check your answer by multiplying the binomials.

Look at another trinomial with the last sign – .

x2 - 4x – 21(x ) (x ) Set up the ( ) and x’s.

(x + ) (x - ) Determine the signs. Since the product is – 21, we need to use two different signs. Put - and + in the ( ).

(x - 7)(x + 3) Find 2 numbers which multiply to give 21 and subtract to give 4.

7 and 3 work (We will need – 7 and + 3 to get – 4.)

Check your answer by multiplying the binomials.

Try factoring these trinomials. Then check your answers.

1. x2 + 11x + 28

2. X2 + 8x + 15

3. X2 – 9x + 20

4. X2 – 7x + 6

5. X2 + 4x – 45

6. X2 – 6x – 16

7. X2 + 2x – 24

1. (x + 7)(x + 4)

2. (x + 5)(x + 3)

3. (x – 5)(x – 4)

4. (x – 6)(x – 1)

5. (x + 9)(x – 5)

6. (x – 8)(x + 2)

7. (x + 6)(x – 4)