1. To factor a trinomial of the form: x2 + bx + c
Remember that factoring is the reverse of multiplication and the Distributive Property. ab + ac = a(b + c)
Question: What kind of multiplication gives the product x2+bx+c.
Answer: Two binomial factors that have been multiplied using the FOIL method.
Let’s look at 4 basic product types with a trinomial of: 1.
sum of 2 and 8
product of 2 and 8 2.
sum of -2 and -8
product of -2 and -8
Note: When the signs in the binomials are the same:
1. the constant (third term) is positive (+).
2. the middle term has the common binomial sign.
3. the coefficient of the middle term is the sum of the binomial constants.

2. sum of -2 and 8
product of -2 and 8
sum of 2 and -8
product of 2 and -8
Note: When the signs in the binomials are different:
1. the constant term is negative (–).
2. the middle term has the sign of the larger of the absolute values of the two binomial constants.
3. the coefficient of the middle term is the sum of the binomial constants.
Using these product types (1-4) gives the following factoring patterns.
(# represents the binomial constant)
Trinomial Factored

3. Step 1. Write two sets of parentheses and only place the variable in
Step 1. Write two sets of parentheses and only place the variable in the first positions of each set, i.e. (x )(x ).
Step 2. Make a list of any two factors which multiply to equal c.
Step 3. If the sign in front of c (or the last sign) is
+ : Find which pair of factors from Step 2 will add to equal b
– : Find which pair of factors from Step 2 will subtract to equal b
Step 4. Place the pair of numbers in the last position of each set of parenthesis i.e. (x #1)(x #2)
Note: The order of the numbers does not matter.
Step 5. To put the signs in will take a careful analysis.
If the last sign is a +, both signs are the same. This will depend on the middle term. If the middle term is positive, both signs will be positive. If the middle term is negative both signs will be negative.
If the last sign is a –, one of the signs will be negative and the other will be positive. The larger of the two numbers will have the same sign as the middle term, and the smaller of the two numbers will have the opposite sign of the middle term.
Step 6. Check using the FOIL method.

4. Solution:
(x ) (x ) + 3 8
Write the parentheses and insert the variable in the first positions of each. 24
2. Make a list of all pairs of factors of the constant term.
3. Since the sign of the constant term is positive, we want the pair whose sum is equal to 11. If the last term was negative, we would want the pair whose difference is equal to the coefficient of the middle term.
= +11
6. Check by FOIL
4. Place the numbers into the last position of each parenthesis (order does not matter). 
5. Since the last sign is positive, both signs are the same. Since the middle sign is also positive, both signs are positive.
Answer:
Your Turn Problem #1
Answer:

5. Solution:
(x ) (x ) – + 9 7
Write the parentheses and insert the variable in the first positions of each. 63
2. Make a list of all factors of the constant term.
3. Since the last sign is a negative, we want the pair which subtracts to equal the middle term.
+9 – 7 = +2
4. Place numbers into the last position of each parentheses (order does not matter).
6. Check by FOIL 
Answer:
Your Turn Problem #2
5. Since the last sign is a negative, the signs of the binomials are different. The larger number gets the middle sign, which is positive.

6. Note: The trinomials given in the previous examples were all in descending order. Also, the coefficient of the x2 was positive. If the trinomial is not given in descending order or the coefficient in front of the x2 is not positive, then the first step is to rearrange into descending order and then factor out a –1 if necessary.
Solution: First, rearrange into descending order.
Second, factor out a negative.
Now the trinomial can be completely factored.
Your Turn Problem #3

7. Observe the pattern of the product two binomials, each with two variables.
First and last terms have the variables squared while the middle term has the product of each variable.
If the trinomial to be factored is in the form: ,
write the two sets of parenthesis with the x in the first positions and the y in the last positions. Then factor as in previous examples.
Solution:
Write the two sets of parentheses with the variables in the first and last positions. 32 – + 4 8
Find the factors of 32.
Which pair will subtract to equal the middle term? 4 and 8.
The larger number will get the middle sign which is negative.
Answer:
Your Turn Problem #4

8. 1. Multiply a and c. List the factors.
Not all trinomials can be factored. If this is the case, the answer to be written is “The trinomial is not factorable.”
1. Multiply a and c. List the factors.
2. Which pair will subtract to equal 8?
Answer: None?
Therefore, the trinomial is not factorable.
Your Turn Problem #5
The End.
B.R