Chapter 7 Rational Functions - College of the Redwoods

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662 Chapter 7 Rational Functions However, this approach is not the most efficient way to proceed, as multiplying numerators 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 immediately, 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 “something 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.” 12 Thus, we have two choices when multiplying rational expressions: • Multiply numerators and denominators, factor, then cancel. • Factor numerators and denominators, cancel, then multiply numerators and denominators. It is the latter approach that we will use in this section. example. Let’s look at another 12 Students will sometimes use the phrase “cross-cancel” when working with the product of rational expressions. Unfortunately, this term implies that cancellation can occur only in a diagonal direction, which is far from the truth. We like to tell our students that there is no such term as “cross-cancel.” There is only “cancel,” and the rule is: cancel something on the top for something on the bottom, which is vernacular for “cancel a factor from any numerator and the identical factor from any denominator.” Version: Fall 2007
Section 7.4 Products and Quotients of Rational Functions 663 ◮ Example 6. State restrictions. Simplify the expression x 2 − x − 6 x 2 + 2x − 15 · x2 − x − 30 x 2 − 2x − 8 (7) Use the ac-test to factor each numerator and denominator. Then cancel as shown. x 2 − x − 6 x 2 + 2x − 15 · x2 − x − 30 (x + 2)(x − 3) (x + 5)(x − 6) x 2 = · − 2x − 8 (x − 3)(x + 5) (x + 2)(x − 4) (x + 2)(x − 3) (x + 5)(x − 6) = · (x − 3)(x + 5) (x + 2)(x − 4) = x − 6 x − 4 The first fraction’s denominator has factors x − 3 and x + 5. Hence, x = 3 or x = −5 will make this denominator zero. Therefore, the 3 and −5 are restrictions. The second fraction’s denominator has factors x + 2 and x − 4. Hence, x = −2 or x = 4 will make this denominator zero. Therefore, −2 and 4 are restrictions. of Therefore, for all values of x, except the restrictions −5, −2, 3, and 4, the left side is identical to its right side. x 2 − x − 6 x 2 + 2x − 15 · x2 − x − 30 x 2 − 2x − 8 = x − 6 x − 4 It’s possible to use your graphing calculator to check your results. First, load the left- and right-hand sides of equation (8) into the calculator’s into Y1 and Y2 in your graphing calculator’s Y= menu, as shown in Figure 1(a). Press 2nd TBLSET and set TblStart = −6 and ∆Tbl = 1, as shown in Figure 1(b). Make sure that AUTO is highlighted and selected with the ENTER key on both the independent and dependent variables. Press 2nd TABLE to produce the tabular display in Figure 1(c). (8) (a) (b) (c) (d) Figure 1. Using the table features of the graphing calculator to check the result in equation (8). Remember that the left- and right-hand sides of equation (8) are loaded in Y1 and Y2, respectively. Version: Fall 2007
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7 Rational Functions In this chapte
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Section 7.1 Introducing Rational Fu
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Section 7.9 Index 747 7.9 Index a a
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Chapter 7 Rational Functions - College of the Redwoods

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