Section 7.4
Products and Quotients of Rational Functions
661
Version: Fall 2007
7.4
Products and Quotients of Rational Functions
In this section we deal with products and quotients of rational expressions. Before we
begin, we’ll need to establish some fundamental definitions and technique. We begin
with the definition of the product of two rational numbers.
Definition 1.
Let
a/b
and
c/d
be rational numbers.
The product of these
rational numbers is defined by
a
b
×
c
d
=
a
×
c
b
×
d
,
or more compactly,
a
b
·
c
d
=
ac
bd
.
(2)
The definition simply states that you should multiply the numerators of each ra
tional number to obtain the numerator of the product, and you also multiply the
denominators of each rational number to obtain the denominator of the product. For
example,
2
3
·
5
7
=
2
·
5
3
·
7
=
10
21
.
Of course, you should also check to make sure your final answer is reduced to lowest
terms.
Let’s look at an example.
l⚏
Example 3.
Simplify the product of rational numbers
6
231
·
35
10
.
(4)
First, multiply numerators and denominators together as follows.
6
231
·
35
10
=
6
·
35
231
·
10
=
210
2310
.
However, the answer is not reduced to lowest terms. We can express the numerator as
a product of primes.
210 = 21
·
10 = 3
·
7
·
2
·
5 = 2
·
3
·
5
·
7
It’s not necessary to arrange the factors in ascending order, but every little bit helps.
The denominator can also be expressed as a product of primes.
2310 = 10
·
231 = 2
·
5
·
7
·
33 = 2
·
3
·
5
·
7
·
11
We can now cancel common factors.
210
2310
=
2
·
3
·
5
·
7
2
·
3
·
5
·
7
·
11
=
2
·
3
·
5
·
7
2
·
3
·
5
·
7
·
11
=
1
11
(5)
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662
Chapter 7
Rational Functions
Version: Fall 2007
However, this approach is not the most efficient way to proceed, as multiplying numer
ators and denominators allows the products to grow to larger numbers, as in 210/2310.
It is then a little bit harder to prime factor the larger numbers.
A better approach is to factor the smaller numerators and denominators immedi
ately, as follows.
6
231
·
35
10
=
2
·
3
3
·
7
·
11
·
5
·
7
2
·
5
We could now multiply numerators and denominators, then cancel common factors,
which would match identically the last computation in
equation (5)
.
However, we can also employ the following cancellation rule.
Cancellation Rule.
When working with the product of two or more rational
expressions, factor all numerators and denominators, then cancel. The cancellation
rule is simple: cancel a factor “on the top” for an identical factor “on the bottom.”
Speaking more technically, cancel any factor in any numerator for an identical
factor in any denominator.
Thus, we can finish our computation by canceling common factors, canceling “some
thing on the top for something on the bottom.”
6
231
·
35
10
=
2
·
3
3
·
7
·
11
·
5
·
7
2
·
5
=
2
·
3
3
·
7
·
11
·
5
·
7
2
·
5
=
1
11
Note that we canceled a 2, 3, 5, and a 7 “on the top” for a 2, 3, 5, and 7 “on the
bottom.”
2
Thus, we have two choices when multiplying rational expressions:
•
Multiply numerators and denominators, factor, then cancel.
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