Section 7.2
Reducing Rational Functions
619
Version: Fall 2007
7.2
Reducing Rational Functions
The goal of this section is to learn how to reduce a rational expression to “lowest terms.”
Of course, that means that we will have to understand what is meant by the phrase
“lowest terms.” With that thought in mind, we begin with a discussion of the
greatest
common divisor
of a pair of integers.
First, we define what we mean by “divisibility.”
Definition 1.
Suppose that we have a pair of integers
a
and
b
. We say that “
a
is a divisor of
b
,” or “
a
divides
b
” if and only if there is another integer
k
so that
b
=
ak
. Another way of saying the same thing is to say that
a
divides
b
if, upon
dividing
b
by
a
, the remainder is zero.
Let’s look at an example.
l⚏
Example 2.
What are the divisors of 12?
Because
12 = 1
×
12
, both 1 and 12 are divisors
2
of 12. Because
12 = 2
×
6
, both 2
and 6 are divisors of 12. Finally, because
12 = 3
×
4
, both 3 and 4 are divisors of 12.
If we list them in ascending order, the divisors of 12 are
1
,
2
,
3
,
4
,
6
,
and
12
.
Let’s look at another example.
l⚏
Example 3.
What are the divisors of 18?
Because
18 = 1
×
18
, both 1 and 18 are divisors of 18. Similarly,
18 = 2
×
9
and
18 = 3
×
6
, so in ascending order, the divisors of 18 are
1
,
2
,
3
,
6
,
9
,
and
18
.
The
greatest common divisor
of two or more integers is the largest divisor the
integers share in common. An example should make this clear.
l⚏
Example 4.
What is the greatest common divisor of 12 and 18?
In
Example 2
and
Example 3
, we saw the following.
Divisors of 12
:
1
,
2
,
3
,
4
,
6
,
12
Divisors of 18
:
1
,
2
,
3
,
6
,
9
,
18
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1
The word “divisor” and the word “factor” are synonymous.
2