Chapter 7 Rational Functions - College of the Redwoods

Section 7.4 Products and Quotients of Rational Functions 661
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 rational
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.
◮ 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)
11
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Version: Fall 2007