¡
43
r
2
+
12
r
¡
35
24) 16
x
3
+
44
x
2
+
44
x
+
40
25) 12
n
3
¡
20
n
2
+
38
n
¡
20
26)
8
b
3
¡
4
b
2
¡
4
b
¡
12
27) 36
x
3
¡
24
x
2
y
+ 3
xy
2
+
12
y
3
28) 21
m
3
+ 4
m
2
n
¡
8
n
3
29) 48
n
4
¡
16
n
3
+
64
n
2
¡
6
n
+
36
30) 14
a
4
+
30
a
3
¡
13
a
2
¡
12
a
+ 3
31) 15
k
4
+
24
k
3
+
48
k
2
+
27
k
+
18
32) 42
u
4
+
76
u
3
v
+
17
u
2
v
2
¡
18
v
4
33) 18
x
2
¡
15
x
¡
12
34) 10
x
2
¡
55
x
+
60
35) 24
x
2
¡
18
x
¡
15
36) 16
x
2
¡
44
x
¡
12
37)
7
x
2
¡
49
x
+
70
38) 40
x
2
¡
10
x
¡
5
39) 96
x
2
¡
6
40) 36
x
2
+
108
x
+
81
107

Section 3.6: Special Products
Objective: Recognize and use special product rules of a sum and a dif-
ference and perfect squares to multiply polynomials.
There are a few shortcuts that we can take when multiplying polynomials. If we
can recognize them, the shortcuts can help us arrive at the solution much faster.
These shortcuts will also be useful to us as our study of algebra continues.
The ±rst shortcut is often called a
sum and a di±erence
. A sum and a di²er-
ence is easily recognized as the numbers and variables are exactly the same, but
the sign in the middle is different (one sum, one difference). To illustrate the
shortcut, consider the following example, where we multiply using the distributing
method.
Example 1.
Simplify.
(
a
+
b
)(
a
¡
b
)
Distribute
(
a
+
b
)
a
(
a
+
b
)
¡
b
(
a
+
b
)
Distribute
a
and
¡
b
a
2
+
ab
¡
ab
¡
b
2
Combine like terms
ab
¡
ab
a
2
¡
b
2
Our Solution
The important part of this example is that the middle terms subtracted to zero.
Rather than going through all this work, when we have a sum and a di²erence, we
will jump right to our solution by squaring the ±rst term and squaring the last
term, putting a subtraction between them. This is illustrated in the following
example.
Example 2.
Simplify.
(
x
¡
5)(
x
+ 5)
Recognize sum and di²erence
Square both
x
and
5;
put subtraction between the squares
x
2
¡
25
Our Solution
This is
much quicker than going through the work of multiplying and combining
like terms. Often students ask if they can just multiply out using another method
and not learn the shortcut. These shortcuts are going to be very useful when we
get to factoring polynomials, or reversing the multiplication process.
For this
reason it is very important to be able to recognize these shortcuts. More examples
are shown below.
108

Example 3.
Simplify.
(3
x
+ 7)(3
x
¡
7)
Recognize sum and di²erence
Square both
3
x
and
7;
put subtraction between the squares
9
x
2
¡
49
Our Solution
Example 4.
Simplify.
(2
x
¡
6
y
)(2
x
+ 6
y
)
Recognize sum and di²erence
Square both
2
x
and
6
y
;
put subtraction between the squares
4
x
2
¡
36
y
2
Our Solution
It is interesting to note that while we can multiply and get an answer like
a
2
¡
b
2
(with subtraction), it is impossible to multiply binomial expressions and end up
with a product such as
a
2
+
b
2
(with addition).
There is also a shortcut to multiply a
perfect square
, which is a binomial raised
to the power two. The following example illustrates multiplying a perfect square.

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- Spring '14
- Abdelaziz
- Math, Algebra, Polynomials, Exponents, Derivative, Product Rule, Exponentiation, Exponent properties