0.4
Factoring Polynomials
41
■
Answer: a.
(
x
2)(
x
2
2
x
4)
b.
(
x
4)(
x
2
4
x
16)
Factoring a Trinomial as a Product
of Two Binomials
The first step in factoring is to look for a common factor. If there is no common factor, look
to see whether the polynomial is a special form for which we know the factoring formula.
If it is not of such a special form and if it is a trinomial, then we proceed with a general
factoring strategy.
We know that (
x
3)(
x
2)
x
2
5
x
6, so we say the
factors
of
x
2
5
x
6 are
(
x
3)
and
(
x
2)
. In factored form we have
x
2
5
x
6
(
x
3)(
x
2). Recall the
FOIL method from Section 0.3. The product of the last terms (3 and 2) is 6, and the sum of
the products of the inner terms (3
x
) and the outer terms (2
x
) is 5
x
. Let’s pretend for a minute
that we didn’t know this factored form but had to work with the general form:
The goal is to find
a
and
b
. We start by multiplying the two binomials on the right.
Compare the expression we started with on the left with the expression on the far right
x
2
5
x
6
x
2
(
a
b
)
x
ab
. We see that
ab
6
and
(
a
b
)
5
.
Start
with the
possible combinations of
a
and
b
whose product is 6, and
then
look among those for
the combination whose sum is 5.
ab
6
a
,
b
:
1, 6
1,
6
2, 3
2,
3
a
b
7
7
5
5
All of the possible
a
,
b
combinations in the first row have a product equal to 6, but only one
of those has a sum equal to 5. Therefore the factored form is
x
2
+
5
x
+
6
=
(
x
+
a
)(
x
+
b
)
=
(
x
+
2)(
x
+
3)
x
2
+
5
x
+
6
=
(
x
+
a
)(
x
+
b
)
=
x
2
+
ax
+
bx
+
ab
=
x
2
+
(
a
+
b
)
x
+
ab
x
2
+
5
x
+
6
=
(
x
+
a
)(
x
+
b
)
EXAMPLE 5
Factoring the Difference of Two Cubes
Factor
x
3
125.
Solution:
Rewrite as the difference of two cubes.
x
3
125
x
3
5
3
Write the difference of two cubes formula.
a
3
b
3
(
a
b
)(
a
2
ab
b
2
)
Let
a
x
and
b
5.
x
3
125
x
3
5
3
(
x
5)(
x
2
5
x
25)
■
YOUR TURN
Factor:
a.
x
3
8
b.
x
3
64

42
CHAPTER 0
Prerequisites and Review
■
Answer:
(
x
4)(
x
5)
■
Answer:
(
x
6)(
x
3)
In Example 6, all terms in the trinomial are positive. When the constant term is negative,
then (regardless of whether the middle term is positive or negative) the factors will be
opposite in sign, as illustrated in Example 7.
EXAMPLE 7
Factoring a Trinomial
Factor
x
2
3
x
28.
Solution:
Write the trinomial as a product of two
binomials in general form.
x
2
3
x
28
(
x
)(
x
)
Write all of the integers whose product is
28.
Integers whose product is
28
1,
28
1, 28
2,
14
2, 14
4,
7
4, 7
Determine the sum of the integers.
Integers whose product is
28
1,
28
1, 28
2,
14
2, 14
4,
7
4, 7
Sum
27
27
12
12
3
3
Select 4,
7 because the product is
28
(last term of the trinomial) and the sum is
3
(middle term coefficient of the trinomial).
x
2
3
x
28
(
x
4)(
x
7)
Check:
(
x
4)(
x
7)
x
2
7
x
4
x
28
x
2
3
x
28
✓
■
YOUR TURN
Factor
x
2
3
x
18.
n
n
EXAMPLE 6
Factoring a Trinomial
Factor
x
2
10
x
9.
Solution:
Write the trinomial as a product of two
binomials in general form.
x
2
10
x
9
(
x
n
)(
x
n
)
Write all of the integers whose product is 9.

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