The middle term of the trinomial is
x
, so we look for the sum of the integers that equals
1. Since no sum exists for the given combinations, we say that this polynomial is
prime
(irreducible) over the integers.
Determine the sum
of the integers.
Integers whose product is
8
1,
8
1, 8
4,
2
4, 2
Integers whose product is
8
1,
8
1, 8
4,
2
4, 2
Sum
7
7
2
2
Study Tip
In Example 9, Step 3, we can
eliminate any factors that have a
common factor since there is no
common factor to the terms in the
trinomial.

0.4
Factoring Polynomials
45
1.
Factor out the greatest common factor (monomial).
2.Identify any special polynomial forms and apply factoring formulas.
3.
Factor a trinomial into a product of two binomials: (
ax
b
)(
cx
d
).
4.
Factor by grouping.
S
TRATEGY FOR FACTORING POLYNOMIALS
Study Tip
When factoring, always start by
factoring out the GCF.
A Strategy for Factoring Polynomials
The first step in factoring a polynomial is to look for the greatest common factor. When
specifically factoring trinomials, look for special known forms: a perfect square or a
difference of two squares. A general approach to factoring a trinomial uses the FOIL
method in reverse. Finally, we look for factoring by grouping. The following strategy for
factoring polynomials is based on the techniques discussed in this section.
EXAMPLE 12
Factoring a Polynomial by Grouping
Factor 2
x
2
2
x
x
1.
Solution:
Group the terms that have a common factor.
(2
x
2
2
x
)
(
x
1)
Factor out the common factor in each pair of parentheses.
2
x
(
x
1)
1(
x
1)
Use the distributive property.
(2
x
1)(
x
1)
■
YOUR TURN
Factor
x
3
x
2
3
x
3.
EXAMPLE 13
Factoring Polynomials
Factor:
a.
3
x
2
6
x
3
b.
4
x
3
2
x
2
6
x
c.
15
x
2
7
x
2
d.
x
3
x
2
x
2
2
Solution (a):
Factor out the greatest common factor.
3
x
2
6
x
3
3(
x
2
2
x
1)
The trinomial is a perfect square.
3(
x
1)
2
Solution (b):
Factor out the greatest common factor.
4
x
3
2
x
2
6
x
2
x
(2
x
2
x
3)
Use the FOIL method in reverse to
factor the trinomial.
2
x
(2
x
3)(
x
1)
Solution (c):
There is no common factor.
15
x
2
7
x
2
Use the FOIL method in reverse to factor the trinomial.
(3
x
2)(5
x
1)
Solution (d):
Factor by grouping.
x
3
x
2
x
2
2
(
x
3
x
)
(2
x
2
2)
x
(
x
2
1)
2(
x
2
1)
(
x
2)(
x
2
1)
Factor the difference of two squares.
(
x
2)(
x
1)(
x
1)
■
Answer:
(
x
1)(
x
2
3)

46
CHAPTER 0
Prerequisites and Review
Factoring a Trinomial as a Product of Two Binomials
x
2
bx
c
(
x
?)(
x
?)
1.
Find all possible combinations of factors whose product
is
c
.
2.
Of the combinations in Step 1, look for the sum of factors
that equals
b
.
ax
2
bx
c
(?
x
?)(?
x
?)
1.
Find all possible combinations of the first terms whose
product is
ax
2
.
2.
Find all possible combinations of the last terms whose
product is
c
.

#### You've reached the end of your free preview.

Want to read all 10 pages?

- Summer '17
- juan alberto
- Fractions, Elementary arithmetic, Greatest common divisor