After studying this lesson, you will be able to:

• Factor trinomials.

Steps of Factoring:

1. Factor out the GCF

2. Look at the number of terms:

• 2 Terms: Look for the Difference of 2 Squares
• 3 Terms: Factor the Trinomial
• 4 Terms: Factor by Grouping

3. Factor Completely

4. Check by Multiplying

This lesson will concentrate on the second step of factoring:Factoring Trinomials.

**When there are 3 terms, we are factoring trinomials. Don'tforget to look for a GCF first.**

Factoring trinomials often requires some trial and error.Don't get frustrated. Try all possible combinations.

Rules for determining the signs in each factor:

If the Constant Term is Positive, both signs will bethe same (this means that either both will be positiveor both will be negative)

OR

If the Constant Term is Negative, the signs will bedifferent (this means that one will be positive and onewill be negative)

OR

Example 1

Factor x ² + 7xy + 10y ²

This is a trinomial (has 3 terms). There is no GCF other thanone. So, we start with 2 parentheses:

Using our signs rules, we can determine the signs for thefactors. Since the constant term is positive we know the signswill be the same. Since we want the factors to add up to +7xy thesigns will both have to be positive. Keep this in mind.

1 st : Find the factors of the first term. The factors of x² are x and x. These go in the first positions. We can alsogo ahead and put in the signs (both positive)

2 nd : Find the factors of the constant term. The factors of10y ² are 1y, 10y and 2y, 5y . Remember, we need theinside/outside combination to add up to the middle term which is+7xy. Since 2xy and 5xy add up to 7xy, we choose those factors:

(x + 2y ) ( x + 5y)

Check by using FOIL (x + 2y) (x + 5y) x ² + 5xy + 2xy + 10y ²which is x ² + 7xy + 10y ²

Example 2

Factor x ² - 8x - 20

This is a trinomial (has 3 terms). There is no GCF other thanone. So, we start with 2 parentheses:

Using our signs rules, we can determine the signs for thefactors. Since the constant term is negative we know the signswill be different. We want the factors to add up to -8x. Let'sremember that the signs are different. We aren't sure just yetwhich sign will be positive and which will be negative.

1 st : Find the factors of the first term. The factors of x 2are x and x. These go in the first positions. We don't yet put inthe signs because we aren't sure where they go.

2 nd : Find the factors of the constant term. The factors of20 are 1, 20 and 2, 10 and 4, 5. Remember, we need theinside/outside combination to add up to the middle term which is-8x. We have to remember that we are working with differentsigns. Therefore, we're really looking for a pair of factors thathave a difference of 8. Since 2 and 10 have a difference of 8, wechoose those factors. Now we have to decide where to put thepositive and where to put the negative. The 2 and the 10 have toadd up to -8 so we make the 10 negative and the 2 positive.

(x + 2 ) ( x - 10 )

Check by using FOIL (x + 2) ( x - 10 ) x ² - 10x + 2x - 20 which is x² - 8x - 20

Example 5

Factor x ² + 4x - 12

This is a trinomial (has 3 terms). There is no GCF other thanone. So, we start with 2 parentheses:

Using our signs rules, we can determine the signs for thefactors. Since the constant term is negative we know the signswill be different. We want the factors to add up to +4x. Let'sremember that the signs are different. We aren't sure just yetwhich sign will be positive and which will be negative.

1st : Find the factors of the first term. The factors of x 2are x and x. These go in the first positions. We don't yet put inthe signs because we aren't sure where they go.

2 nd : Find the factors of the constant term. The factors of12 are 1, 12 and 2, 6 and 3, 4. Remember, we need theinside/outside combination to add up to the middle term which is-+4x. We have to remember that we are working with differentsigns. Therefore, we're really looking for a pair of factors thathave a difference of 4. Since 2 and 6 have a difference of 4, wechoose those factors. Now we have to decide where to put thepositive and where to put the negative. The 2 and the 6 have toadd up to +4 so we make the 2 negative and the 6 positive.

(x - 2 ) ( x + 6 ) note : it would be the same thing to writethe factors in this order: (x - 6) (x - 2 )

Check by using FOIL (x - 2) ( x + 6 ) x ² +6x - 2x - 12 which is x² + 4x - 12