678 <strong>Chapter</strong> 7 <strong>Rational</strong> <strong>Functions</strong> Version: Fall 2007
Section 7.5 Sums and Differences <strong>of</strong> <strong>Rational</strong> <strong>Functions</strong> 679 7.5 Sums and Differences <strong>of</strong> <strong>Rational</strong> <strong>Functions</strong> In this section we concentrate on finding sums and differences <strong>of</strong> rational expressions. However, before we begin, we need to review some fundamental ideas and technique. First and foremost is <strong>the</strong> concept <strong>of</strong> <strong>the</strong> multiple <strong>of</strong> an integer. This is best explained with a simple example. The multiples <strong>of</strong> 8 is <strong>the</strong> set <strong>of</strong> integers {8k : k is an integer}. In o<strong>the</strong>r words, if you multiply 8 by 0, ±1, ±2, ±3, ±4, etc., you produce what is known as <strong>the</strong> multiples <strong>of</strong> 8. Multiples <strong>of</strong> 8 are: 0, ±8, ±16, ±24, ±32, etc. However, for our purposes, only <strong>the</strong> positive multiples are <strong>of</strong> interest. So we will say: Multiples <strong>of</strong> 8 are: 8, 16, 24 , 32, 40, 48 , 56, 64, 72 , . . . Similarly, we can list <strong>the</strong> positive multiples <strong>of</strong> 6. Multiples <strong>of</strong> 6 are: 6, 12, 18, 24 , 30, 36, 42, 48 , 54, 60, 66, 72 , . . . We’ve framed those numbers that are multiples <strong>of</strong> both 8 and 6. These are called <strong>the</strong> common multiples <strong>of</strong> 8 and 6. Common multiples <strong>of</strong> 8 and 6 are: 24, 48, 72, . . . The smallest <strong>of</strong> this list <strong>of</strong> common multiples <strong>of</strong> 8 and 6 is called <strong>the</strong> least common multiple <strong>of</strong> 8 and 6. We will use <strong>the</strong> following notation to represent <strong>the</strong> least common multiple <strong>of</strong> 8 and 6: LCM(8, 6). Hopefully, you will now feel comfortable with <strong>the</strong> following definition. Definition 1. Let a and b be integers. The least common multiple <strong>of</strong> a and b, denoted LCM(a, b), is <strong>the</strong> smallest positive multiple that a and b have in common. For larger numbers, listing multiples until you find one in common can be impractical and time consuming. Let’s find <strong>the</strong> least common multiple <strong>of</strong> 8 and 6 a second time, only this time let’s use a different technique. First, write each number as a product <strong>of</strong> primes in exponential form. 8 = 2 3 6 = 2 · 3 14 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ Version: Fall 2007