o Section 5: Sums and Differences of Rational Functions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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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 · 3 Copyrighted material. See: 1
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680 Chapter 7 Rational Functions Version: Fall 2007 Here’s the rule. A Procedure to Find the LCM. To find the LCM of two integers, proceed as follows. 1. Express the prime factorization of each integer in exponential format. 2. To find the least common multiple, write down every prime number that ap- pears, then affix the largest exponent of that prime that appears. In our example, the primes that occur are 2 and 3. The highest power of 2 that occurs is 2 3 . The highest power of 3 that occurs is 3 1 . Thus, the LCM (8 , 6) is LCM (8 , 6) = 2 3 · 3 1 = 24 . Note that this result is identical to the result found above by listing all common mul- tiples and choosing the smallest. Let’s try a harder example. l⚏ Example 2. Find the least common multiple of 24 and 36. Using the first technique, we list the multiples of each number, framing the multiples in common. Multiples of 24: 24 , 48 , 72 , 96 , 120 , 144 , 168 , . . . Multiples of 36: 36 , 72 , 108 , 144 , 180 . . . The multiples in common are 72, 144, etc., and the least common multiple is LCM (24 , 36) = 72 .
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