Chapter 7 Rational Functions - College of the Redwoods
Chapter 7 Rational Functions - College of the Redwoods
Chapter 7 Rational Functions - College of the Redwoods
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Section 7.2 Reducing <strong>Rational</strong> <strong>Functions</strong> 619<br />
7.2 Reducing <strong>Rational</strong> <strong>Functions</strong><br />
The goal <strong>of</strong> this section is to learn how to reduce a rational expression to “lowest terms.”<br />
Of course, that means that we will have to understand what is meant by <strong>the</strong> phrase<br />
“lowest terms.” With that thought in mind, we begin with a discussion <strong>of</strong> <strong>the</strong> greatest<br />
common divisor <strong>of</strong> a pair <strong>of</strong> integers.<br />
First, we define what we mean by “divisibility.”<br />
Definition 1. Suppose that we have a pair <strong>of</strong> integers a and b. We say that “a<br />
is a divisor <strong>of</strong> b,” or “a divides b” if and only if <strong>the</strong>re is ano<strong>the</strong>r integer k so that<br />
b = ak. Ano<strong>the</strong>r way <strong>of</strong> saying <strong>the</strong> same thing is to say that a divides b if, upon<br />
dividing b by a, <strong>the</strong> remainder is zero.<br />
Let’s look at an example.<br />
◮ Example 2. What are <strong>the</strong> divisors <strong>of</strong> 12?<br />
Because 12 = 1 × 12, both 1 and 12 are divisors 6 <strong>of</strong> 12. Because 12 = 2 × 6, both 2<br />
and 6 are divisors <strong>of</strong> 12. Finally, because 12 = 3 × 4, both 3 and 4 are divisors <strong>of</strong> 12.<br />
If we list <strong>the</strong>m in ascending order, <strong>the</strong> divisors <strong>of</strong> 12 are<br />
1, 2, 3, 4, 6, and 12.<br />
Let’s look at ano<strong>the</strong>r example.<br />
◮ Example 3. What are <strong>the</strong> divisors <strong>of</strong> 18?<br />
Because 18 = 1 × 18, both 1 and 18 are divisors <strong>of</strong> 18. Similarly, 18 = 2 × 9 and<br />
18 = 3 × 6, so in ascending order, <strong>the</strong> divisors <strong>of</strong> 18 are<br />
1, 2, 3, 6, 9, and 18.<br />
The greatest common divisor <strong>of</strong> two or more integers is <strong>the</strong> largest divisor <strong>the</strong><br />
integers share in common. An example should make this clear.<br />
◮ Example 4. What is <strong>the</strong> greatest common divisor <strong>of</strong> 12 and 18?<br />
In Example 2 and Example 3, we saw <strong>the</strong> following.<br />
Divisors <strong>of</strong> 12 : 1 , 2 , 3 , 4, 6 , 12<br />
Divisors <strong>of</strong> 18 : 1 , 2 , 3 , 6 , 9, 18<br />
5<br />
Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/<br />
6 The word “divisor” and <strong>the</strong> word “factor” are synonymous.<br />
Version: Fall 2007