7.2 Reducing Rational Functions

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620 Chapter 7 Rational Functions 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. As we saw in Example 4, the greatest common divisor of 12 and 18 is 6. Therefore, the fraction 12/18 is not reduced to lowest terms. However, we can reduce 12/18 to lowest terms by dividing both numerator and denominator by their greatest common divisor. That is, 12 18 = 12 ÷ 6 18 ÷ 6 = 2 3 . Note that GCD(2, 3) = 1, so 2/3 is reduced to lowest terms. Version: Fall 2007
Section 7.2 Reducing Rational Functions 621 When it is difficult to ascertain the greatest common divisor, we’ll find it more efficient to proceed as follows: • Prime factor both numerator and denominator. • Cancel common factors. Thus, to reduce 12/18 to lowest terms, first express both numerator and denominator as a product of prime numbers, then cancel common primes. 12 18 = 2 · 2 · 3 2 · 3 · 3 = 2 · 2 · 3 2 · 3 · 3 = 2 3 When you cancel a 2, you’re actually dividing both numerator and denominator by 2. When you cancel a 3, you’re actually dividing both numerator and denominator by 3. Note that doing both (dividing by 2 and then dividing by 3) is equivalent to dividing both numerator and denominator by 6. We will favor this latter technique, precisely because it is identical to the technique we will use to reduce rational functions to lowest terms. However, this “cancellation” technique has some pitfalls, so let’s take a moment to discuss some common cancellation mistakes. Cancellation You can spark some pretty heated debate amongst mathematics educators by innocently mentioning the word “cancellation.” There seem to be two diametrically opposed camps, those who don’t mind when their students use the technique of cancellation, and on the other side, those that refuse to even use the term “cancellation” in their classes. Both sides of the argument have merit. As we showed in equation (8), we can reduce 12/18 quite efficiently by simply canceling common factors. On the other hand, instructors from the second camp prefer to use the phrase “factor out a 1” instead of the phrase “cancel,” encouraging their students to reduce 12/18 as follows. 12 18 = 2 · 2 · 3 2 · 3 · 3 = 2 3 · 2 · 3 2 · 3 = 2 3 · 1 = 2 3 This is a perfectly valid technique and one that, quite honestly, avoids the quicksand of “cancellation mistakes.” Instructors who grow weary of watching their students “cancel” when they shouldn’t are quite likely to promote this latter technique. However, if we can help our students avoid “cancellation mistakes,” we prefer to allow our students to cancel common factors (as we did in equation (8)) when reducing fractions such as 12/18 to lowest terms. So, with these thoughts in mind, let’s discuss some of the most common cancellation mistakes. Let’s begin with a most important piece of advice. How to Avoid Cancellation Mistakes. You may only cancel factors, not addends. To avoid cancellation mistakes, factor completely before you begin to cancel. (8) Version: Fall 2007
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7.2 Reducing Rational Functions

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