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. In the
previous problems, the first term has not had a coefficient. We
will now look at problems that do have coefficients in the first
term. This adds another level of trial and error or
"guessing".
One thing that will make the "guessing" more
accurate is to look for a prime number in the first term or the
constant term. Remember, a prime number only has 2 factors.....1
and itself. If the coefficient of the first term or the constant
term is prime, start there and "lock in" those factors.
Example 1
Factor 5x 2 - 17x + 14
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 -17x the
signs will both have to be negative. Keep this in mind.
1 st : Since the coefficient of the first term is prime (5),
we will start with the first term. Find the factors of the first
term. The factors of 5x 2 are 1x and 5x. 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
14 are 1,14 and 2, 7. Remember, we need the inside/outside
combination to add up to the middle term which is -17x. This time
we don't just consider the factors of the constant term because
the first term also had factors. Here's where the guessing comes
in. Let's try the factors 2,7 and see what happens.
(1x - 2 ) (5 x - 7 ) Let's check the inside/outside
combination. If we multiply inside, -2 times 5x gives us -10x.
Multiplying outside 1x times -7 gives us -7x. Add up the
inside/outside combination: -10x + -7x is equal to -17x which is
our middle term. We made a lucky guess!! Note: If the 2 and 7
hadn't worked, we should try 7 and 2. If that didn't work either,
we would try 1 and 14 or 14 and 1. The point is, you keep trying
until you find the right combination.
Check by using FOIL (x - 2) (5x - 7) 
5x 2 - 7x - 10x + 14 which is
5x 2 - 17x + 14
Example 2
Factor 8x 2 - 10x + 3
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 -10x the
signs will both have to be negative. Keep this in mind.
1 st : Since the coefficient of the constant term is prime
(3), we will start with the constant term. Find the factors of
the first term. The factors of 3 are 1 and 3. These go in the
last positions. We can also go ahead and put in the signs (both
negative)
2 nd : Find the factors of the first term. The factors of 8x
2 are 1x, 8x and 2x, 4x. Remember, we need the
inside/outside combination to add up to the middle term which is
-10x. This time we don't just consider the factors of the
constant term because the first term also had factors. Here's
where the guessing comes in. Let's try the factors 4x ,2x and see
what happens.
(4x - 1 ) (2x - 3 ) Let's check the inside/outside
combination. If we multiply inside, -1 times 2x gives us -2x.
Multiplying outside 4x times -3 gives us -12x. Add up the
inside/outside combination: -2x + -12x is equal to -14x which is
NOT our middle term. This is not a lucky guess. Ok, let's reverse
the factors 4x, 2x and try it that way:
(2x - 1 ) ( 4x - 3) Let's check the inside/outside
combination. If we multiply inside, -1 times 4x gives us -4x.
Multiplying outside 2x times -3 gives us -6x. Add up the
inside/outside combination: -4x + -6x is equal to -10x which is
our middle term.
Check by using FOIL (2x - 1) (4x - 3) 8x 2 - 6x - 4x + 3 which is 8x
2 - 10x + 3
