0.3
Polynomials: Basic Operations
31
First
Inner
Product of
First
Terms
Product of
Last
Terms
Outer
Last
Product of
Outer
Terms
Product of
Inner
Terms
Special Products
The method outlined for multiplying polynomials works for
all
products of polynomials. For
the special case when both polynomials are binomials, the
FOIL method
can also be used.
W
ORDS
M
ATH
Apply the distributive property.
(5
x
1)(2
x
3)
5
x
(2
x
3)
1(2
x
3)
Apply the distributive property.
5
x
(2
x
)
5
x
(3)
1(2
x
)
1(3)
Multiply each set of monomials.
10
x
2
15
x
2
x
3
Combine like terms.
10
x
2
13
x
3
The FOIL method finds the
products of the
F
irst terms,
O
uter terms,
I
nner terms, and
(5
x
1)(2
x
3)
10
x
2
15
x
2
x
3
L
ast terms.
EXAMPLE 6
Multiplying Binomials Using the FOIL Method
Multiply (3
x
1)(2
x
5) using the FOIL method.
Solution:
Multiply the
first
terms.
(3
x
)(2
x
)
6
x
2
Multiply the
outer
terms.
(3
x
)(
5)
15
x
Multiply the
inner
terms.
(1)(2
x
)
2
x
Multiply the
last
terms.
(1)(
5)
5
Add the first, outer, inner, and last terms,
and identify the like terms.
Combine like terms.
6
x
2
13
x
5
■
YOUR TURN
Multiply (2
x
3)(5
x
2).
(3
x
+
1)(2
x
-
5)
=
6
x
2
-
15
x
+
2
x
-
5
■
Answer:
10
x
2
19
x
6
Study Tip
When the binomials are of the form
(
ax
b
)(
cx
d
), the outer and inner
terms will be like terms and can be
combined.
EXAMPLE 5
Multiplying Two Polynomials
Multiply and simplify
.
Solution:
Multiply each term of the
first trinomial by the
entire second trinomial.
Identify like terms.
Combine like terms.
2
x
4
13
x
3
30
x
2
26
x
7
■
YOUR TURN
Multiply and simplify (
-
x
3
+
2
x
-
4)(3
x
2
-
x
+
5).
=
2
x
4
-
10
x
3
+
14
x
2
-
3
x
3
+
15
x
2
-
21
x
+
x
2
-
5
x
+
7
=
2
x
2
(
x
2
-
5
x
+
7)
-
3
x
(
x
2
-
5
x
+
7)
+
1(
x
2
-
5
x
+
7)
(
x
2
-
5
x
+
7)
(2
x
2
-
3
x
+
1)
■
Answer:
3
x
5
x
4
x
3
14
x
2
14
x
20
EXAMPLE 7
Multiplying Binomials Resulting in Special Products
Find the following:
a.
(
x
5)(
x
5)
b.
(
x
5)
2
c.
(
x
5)
2
Solution:
a.
(
x
-
5)(
x
+
5)
=
x
2
+
5
x
-
5
x
-
5
2
=
x
2
-
5
2
=
x
2
-
25
32
CHAPTER 0
Prerequisites and Review
Square of a binomial sum:
(
a
b
)
2
(
a
b
)(
a
b
)
a
2
2
ab
b
2
Square of a binomial difference:
(
a
b
)
2
(
a
b
)(
a
b
)
a
2
2
ab
b
2
P
ERFECT SQUARES
(
a
+
b
)(
a
-
b
)
=
a
2
-
b
2
D
IFFERENCE OF TWO SQUARES
EXAMPLE 8
Finding the Square of a Binomial Sum
Find (
x
3)
2
.
INCORRECT
ERROR:
Don’t forget the middle term, which is
twice the product of the two terms in
the binomial.
(
x
+
3)
2
Z
x
2
+
9
CORRECT
(
x
3)
2
(
x
3)(
x
3)
x
2
3
x
3
x
9
x
2
6
x
9
Forgetting the middle term, which is
twice
the product of the two terms in
the binomial.
C
O M M O N
M
I S TA K E
★
Let
a
and
b
be any real number, variable, or algebraic expression in the following
special products.
Some products of binomials occur frequently in algebra and are given special names.
Example 7 illustrates the
difference of two squares
and
perfect squares
.
r
First
r
Inner
s
Difference of two squares
r
Outer
r
Last
r
First
r
Inner
r
Outer
r
First
r
Inner
r
Outer
r
Last
r
Last
b.
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- Summer '17
- juan alberto
- Algebra, Distributive Property, GCF
