Section 7.5
Sums and Differences of Rational Functions
679
Version: Fall 2007
7.5
Sums and Differences of Rational Functions
In this section we concentrate on finding sums and differences of rational expressions.
However, before we begin, we need to review some fundamental ideas and technique.
First and foremost is the concept of the multiple of an integer. This is best explained
with a simple example. The multiples of 8 is the set of integers
{
8
k
:
k
is an integer
}
.
In other words, if you multiply 8 by 0,
±
1
,
±
2
,
±
3
,
±
4
, etc., you produce what is
known as the multiples of 8.
Multiples of 8 are:
0
,
±
8
,
±
16
,
±
24
,
±
32
,
etc.
However, for our purposes, only the positive multiples are of interest. So we will say:
Multiples of 8 are:
8
,
16
,
24
,
32
,
40
,
48
,
56
,
64
,
72
, . . .
Similarly, we can list the positive multiples of 6.
Multiples of 6 are:
6
,
12
,
18
,
24
,
30
,
36
,
42
,
48
,
54
,
60
,
66
,
72
, . . .
We’ve framed those numbers that are multiples of both 8 and 6. These are called the
common multiples
of 8 and 6.
Common multiples of 8 and 6 are:
24
,
48
,
72
, . . .
The smallest of this list of common multiples of 8 and 6 is called the
least common
multiple
of 8 and 6. We will use the following notation to represent the least common
multiple of 8 and 6: LCM
(8
,
6)
.
Hopefully, you will now feel comfortable with the following definition.
Definition 1.
Let
a
and
b
be integers.
The
least common multiple
of
a
and
b
, denoted LCM
(
a, b
)
, is the smallest positive multiple that
a
and
b
have in
common.
For larger numbers, listing multiples until you find one in common can be imprac-
tical and time consuming. Let’s find the least common multiple of 8 and 6 a second
time, only this time let’s use a different technique.
First, write each number as a product of primes in exponential form.
8 = 2
3
6 = 2
·
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