Factoring Trinomials
After studying this lesson, you will be able to:
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't
forget 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 factor
other than 1
2. Look at the number of terms  it has 3, so it is a
trinomial
3. To factor a trinomial, create 2 sets of parentheses
4. Determine what the factors of the first term are and write
them in the first positions of each parenthesis.
In our example, the
factors of x^{ 2} 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 middle
term.
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 those
factors 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 FOIL
to get x^{ 2} + 7x + 10
If we have some idea what signs to use, that makes our
factoring much easier.
Rules for determining the signs in each factor:
If the Constant Term is Positive, both signs will be
the same (this means that either both will be positive
or both will be negative)
OR
If the Constant Term is Negative, the signs will be
different (this means that one will be positive and one
will be negative)
OR
Example 1
Factor x^{ 2} + 5x + 6
This is a trinomial (has 3 terms). There is no GCF other than
one. So, we start with 2 parentheses:
Using our signs rules, we can determine the signs for the
factors. Since the constant term is positive we know the signs
will be the same. Since we want the factors to add up to +5x the
signs will both have to be positive. Keep this in mind.
1 st : Find the factors of the first term. The factors of x^{
2} are x and x. These go in the first positions. We can also
go ahead and put in the signs (both positive)
2 nd : Find the factors of the constant term. The factors of 6
are 1, 6 and 2, 3. Remember, we need the inside/outside
combination to add up to the middle term which is 5x. Since 2 and
3 add up to 5, we choose those factors:
(x + 2 ) ( x + 3 )
Check by using FOIL (x + 2) (x + 3) x^{ 2} + 3x + 2x + 6 which is x^{
2} + 5x + 6
Example 2
Factor x^{ 2}  8x + 12
This is a trinomial (has 3 terms). There is no GCF other than
one. So, we start with 2 parentheses:
Using our signs rules, we can determine the signs for the
factors. Since the constant term is positive we know the signs
will be the same. Since we want the factors to add up to 8x the
signs will both have to be negative. Keep this in mind.
1 st : Find the factors of the first term. The factors of x^{
2} are x and x. These go in the first positions. We can also
go ahead and put in the signs (both negative)
2 nd : Find the factors of the constant term. The factors of
12 are 1, 12 and 2, 6 and 3, 4. Remember, we need the
inside/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) x^{ 2}  6x  2x + 12 which is x^{
2}  8x + 12
