Section 4: Products and Quotients of Rational Functions

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

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Section 7.4 Products and Quotients of Rational Functions 661 Version: Fall 2007 7.4 Products and Quotients of Rational Functions In this section we deal with products and quotients of rational expressions. Before we begin, we’ll need to establish some fundamental definitions and technique. We begin with the definition of the product of two rational numbers. Definition 1. Let a/b and c/d be rational numbers. The product of these rational numbers is defined by a b × c d = a × c b × d , or more compactly, a b · c d = ac bd . (2) The definition simply states that you should multiply the numerators of each ra- tional number to obtain the numerator of the product, and you also multiply the denominators of each rational number to obtain the denominator of the product. For example, 2 3 · 5 7 = 2 · 5 3 · 7 = 10 21 . Of course, you should also check to make sure your final answer is reduced to lowest terms. Let’s look at an example. l⚏ Example 3. Simplify the product of rational numbers 6 231 · 35 10 . (4) First, multiply numerators and denominators together as follows. 6 231 · 35 10 = 6 · 35 231 · 10 = 210 2310 . However, the answer is not reduced to lowest terms. We can express the numerator as a product of primes. 210 = 21 · 10 = 3 · 7 · 2 · 5 = 2 · 3 · 5 · 7 It’s not necessary to arrange the factors in ascending order, but every little bit helps. The denominator can also be expressed as a product of primes. 2310 = 10 · 231 = 2 · 5 · 7 · 33 = 2 · 3 · 5 · 7 · 11 We can now cancel common factors. 210 2310 = 2 · 3 · 5 · 7 2 · 3 · 5 · 7 · 11 = 2 · 3 · 5 · 7 2 · 3 · 5 · 7 · 11 = 1 11 (5) Copyrighted material. See: 1
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662 Chapter 7 Rational Functions Version: Fall 2007 However, this approach is not the most efficient way to proceed, as multiplying numer- ators and denominators allows the products to grow to larger numbers, as in 210/2310. It is then a little bit harder to prime factor the larger numbers. A better approach is to factor the smaller numerators and denominators immedi- ately, as follows. 6 231 · 35 10 = 2 · 3 3 · 7 · 11 · 5 · 7 2 · 5 We could now multiply numerators and denominators, then cancel common factors, which would match identically the last computation in equation (5) . However, we can also employ the following cancellation rule. Cancellation Rule. When working with the product of two or more rational expressions, factor all numerators and denominators, then cancel. The cancellation rule is simple: cancel a factor “on the top” for an identical factor “on the bottom.” Speaking more technically, cancel any factor in any numerator for an identical factor in any denominator. Thus, we can finish our computation by canceling common factors, canceling “some- thing on the top for something on the bottom.” 6 231 · 35 10 = 2 · 3 3 · 7 · 11 · 5 · 7 2 · 5 = 2 · 3 3 · 7 · 11 · 5 · 7 2 · 5 = 1 11 Note that we canceled a 2, 3, 5, and a 7 “on the top” for a 2, 3, 5, and 7 “on the bottom.” 2 Thus, we have two choices when multiplying rational expressions: Multiply numerators and denominators, factor, then cancel.
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