Section P.6
Factoring Trinomials
61
Factoring Trinomials of the Form
Try covering the factored forms in the left-hand column below. Can you deter-
mine the factored forms from the trinomial forms?
Factored Form
F
O
I
L
Trinomial Form
Your goal here is to factor trinomials of the form
To begin, consider
the factorization
By multiplying the right-hand side, you obtain the following result.
Sum of
Product of
terms
terms
So, to
factor
a trinomial
into a product of two binomials, you must
find two factors of
c
with a sum of
b
.
Example 1
Factoring Trinomials
Factor the trinomials (a)
and (b)
Solution
(a)
You need to find two factors whose product is
and whose sum is
The product of
and 2 is
The sum of
and 2 is
(b)
You need to find two factors whose product is 6 and whose sum is
The product of
and
is 6
The sum of
and
is
Now try Exercise 7.
Note that when the constant term of the trinomial is positive, its factors must
have
like
signs; otherwise, its factors have
unlike
signs.
5.
2
3
x
2
5
x
6
x
3
x
2
2
3
5.
2.
4
x
2
2
x
8
x
4
x
2
8.
4
2.
8
x
2
5
x
6.
x
2
2
x
8
x
2
bx
c
b
x
c
x
2
x
2
m
n x
mn
x
m
x
n
x
2
nx
mx
mn
x
2
bx
c
x
m
x
n
.
x
2
bx
c
.
3
x
5
x
1
3
x
2
3
x
5
x
5
3
x
2
8
x
5
x
3
x
2
x
2
2
x
3
x
6
x
2
5
x
6
x
1
x
4
x
2
4
x
x
4
x
2
3
x
4
x
2
bx
c
Factoring Trinomials
P.6
•
Factor trinomials of the form
•
Factor trinomials of the form
•
Factor trinomials by grouping
•
Factor perfect square trinomials
•
Select the best factoring
technique using the guidelines
for factoring polynomials
The techniques for factoring
trinomials will help you in solving
quadratic equations.
ax
2
bx
c
x
2
bx
c
What
you should learn:
Why
you should learn it:
Study Tip
Use a list to help you find the
two numbers with the required
product and sum. For Example
1(a):
Factors of
Sum
1,
8
7
2,
4
2
Because
is the required
sum, the correct factorization is
x
2
2
x
8
x
4
x
2 .
2
2,
2
4
1,
7
8
8

62
Chapter P
Prerequisites
When factoring a trinomial of the form
if you have trouble
finding two factors of
c
with a sum of
b
, it may be helpful to list all of the distinct
pairs of factors and then choose the appropriate pair from the list. For instance,
consider the trinomial
For this trinomial,
and
So, you need to find two factors of
with a sum of
as shown at the left. With experience, you will be
able to narrow this list down
mentally
to only two or three possibilities whose
sums can then be tested to determine the correct factorization, which is
Example 2
Factoring a Trinomial
Factor the trinomial
Solution
To factor this trinomial, you need to find two factors whose product is
and
whose sum is
The product of 2 and
is
The sum of 2 and
is
Now try Exercise 11.
Applications of algebra sometimes involve trinomials that have a common
monomial factor. To factor such trinomials completely, first factor out the
common monomial factor. Then try to factor the resulting trinomial by the
methods given in this section.

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- Algebra, Integer factorization, bx c, Factor trinomials of the form