Factoring Trinomials


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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.

Here's an explanation on Factoring Trinomials:

Example:

1. Look for the GCF -in this case there's not a common factorother than 1

2. Look at the number of terms - it has 3, so it is atrinomial

3. To factor a trinomial, create 2 sets of parentheses

4. Determine what the factors of the first term are and writethem in the first positions of each parenthesis.

In our example, thefactors of x² are x and x

5. Determine all the possible factors of the constant term.The factors of 10 are 1, 10 and 2, 5

6. The INSIDE / OUTSIDE COMBINATION must add up to the middleterm.

1 and 10 won't add up to 7 (the middle term)

2 and 5 do add up to 7 ( if both are positive) so thosefactors are the ones we use

Write the factors of the constant term in the last positions:

(x + 2) ( x + 5 )	If we multiply the INSIDE part we get 2x
this is the answer	If we multiply the OUTSIDE part we get 5x
	5x + 2x = 7x (the inside/outside combination adds up to the middle term)

We check the answer by multiplying: (x + 2) ( x + 5) Use FOILto get x² + 7x + 10

If we have some idea what signs to use, that makes ourfactoring much easier.

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)

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

Example 1

Factor x² + 5x + 6

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 +5x 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 of 6are 1, 6 and 2, 3. Remember, we need the inside/outsidecombination to add up to the middle term which is 5x. Since 2 and3 add up to 5, we choose those factors:

(x + 2 ) ( x + 3 )

Check by using FOIL (x + 2) (x + 3) x² + 3x + 2x + 6 which is x² + 5x + 6

Example 2

Factor x² - 8x + 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 positive we know the signswill be the same. Since we want the factors to add up to -8x thesigns will both have to be negative. 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 negative)

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-8x. Since 2 and 6 add up to 8, we choose those factors:

(x - 2 ) ( x - 6 )

Check by using FOIL (x - 2) (x - 6) x² - 6x - 2x + 12 which is x² - 8x + 12