Section 2: Reducing Rational Functions

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

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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 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 The word “divisor” and the word “factor” are synonymous. 2
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620 Chapter 7 Rational Functions Version: Fall 2007 We’ve framed the divisors that 12 and 18 have in common. They are 1, 2, 3, and 6. The “greatest” of these “common” divisors is 6. Hence, we say that “the greatest common divisor of 12 and 18 is 6.” Definition 5. The greatest common divisor of two integers a and b is the largest divisor they have in common. We will use the notation GCD ( a, b ) to represent the greatest common divisor of a and b . Thus, as we saw in Example 4 , GCD (12 , 18) = 6 . When the greatest common divisor of a pair of integers is one, we give that pair a special name. Definition 6. Let a and b be integers. If the greatest common divisor of a and b is one, that is, if GCD ( a, b ) = 1 , then we say that a and b are relatively prime . For example: 9 and 12 are not relatively prime because GCD (9 , 12) = 3 . 10 and 15 are not relatively prime because GCD (10 , 15) = 5 . 8 and 21 are relatively prime because GCD (8 , 21) = 1 . We can now define what is meant when we say that a rational number is reduced to lowest terms. Definition 7. A rational number in the form p/q , where p and q are integers, is said to be reduced to lowest terms if and only if GCD ( p, q ) = 1 . That is, p/q is reduced to lowest terms if the greatest common divisor of both numerator and denominator is 1.
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