Chapter 7 Rational Functions - College of the Redwoods

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624 Chapter 7 Rational Functions Similarly, if we insert x = 4 in the left-hand side of equation (15), 2x − 6 x 2 − 7x + 12 = 2(4) − 6 (4) 2 − 7(4) + 12 = 2 0 . Again, division by zero is undefined. The left-hand side of equation (15) is undefined if x = 4, so the result in equation (15) is not valid if x = 4. Note that the right-hand side of equation (15) is also undefined at x = 4. However, the algebraic work we did above guarantees that the left-hand side of equation (15) will be identical to the right-hand side of equation (15) for all other values of x. For example, if we substitute x = 5 into the left-hand side of equation (15), 2x − 6 x 2 − 7x + 12 = 2(5) − 6 (5) 2 − 7(5) + 12 = 4 2 = 2. On the other hand, if we substitute x = 5 into the right-hand side of equation (15), 2 x − 4 = 2 5 − 4 = 2. Hence, both sides of equation (15) are identical when x = 5. In a similar manner, we could check the validity of the identity in equation (15) for all other values of x. You can use the graphing calculator to verify the identity in equation (15). Load the left- and right-hand sides of equation (15) in Y= menu, as shown in Figure 1(a). Press 2nd TBLSET and adjust settings as shown in Figure 1(b). Be sure that you highlight AUTO for both independent and dependent variables and press ENTER on each to make the selection permanent. In Figure 1(b), note that we’ve set TblStart = 0 and ∆Tbl = 1. Press 2nd TABLE to produce the tabular results shown in Figure 1(c). (a) (b) (c) Figure 1. Using the graphing calculator to check that the left- and right-hand sides of equation (15) are identical. Remember that we placed the left- and right-hand sides of equation (15) in Y1 and Y2, respectively. • In the tabular results of Figure 1(c), note the ERR (error) message in Y1 when x = 3 and x = 4. This agrees with our findings above, where the left-hand side of equation (15) was undefined because of the presence of zero in the denominator when x = 3 or x = 4. • In the tabular results of Figure 1(c), note that the value of Y1 and Y2 agree for all other values of x. Version: Fall 2007
Section 7.2 Reducing Rational Functions 625 We are led to the following key result. Restrictions. In general, when you reduce a rational expression to lowest terms, the expression obtained should be identical to the original expression for all values of the variables in each expression, save those values of the variables that make any denominator equal to zero. This applies to the denominator in the original expression, all intermediate expressions in your work, and the final result. We will refer to any values of the variable that make any denominator equal to zero as restrictions. Let’s look at another example. ◮ Example 16. Reduce the expression to lowest terms. State all restrictions. 2x 2 + 5x − 12 4x 3 + 16x 2 − 9x − 36 (17) The numerator is a quadratic trinomial with ac = (2)(−12) = −24. The integer pair −3 and 8 have product −24 and sum 5. Break the middle term of the polynomial in the numerator into a sum using this integer pair, then factor by grouping. Factor the denominator by grouping. 2x 2 + 5x − 12 = 2x 2 − 3x + 8x − 12 = x(2x − 3) + 4(2x − 3) = (x + 4)(2x − 3) 4x 3 + 16x 2 − 9x − 36 = 4x 2 (x + 4) − 9(x + 4) = (4x 2 − 9)(x + 4) = (2x + 3)(2x − 3)(x + 4) Note how the difference of two squares pattern was used to factor 4x 2 − 9 = (2x + 3)(2x − 3) in the last step. Now that we’ve factored both numerator and denominator, we cancel common factors. 2x 2 + 5x − 12 4x 3 + 16x 2 − 9x − 36 = (x + 4)(2x − 3) (2x + 3)(2x − 3)(x + 4) (x + 4)(2x − 3) = (2x + 3)(2x − 3)(x + 4) = 1 2x + 3 We must now determine the restrictions. This means that we must find those values of x that make any denominator equal to zero. Version: Fall 2007
Page 1 and 2: 7 Rational Functions In this chapte
Page 3 and 4: Section 7.1 Introducing Rational Fu
Page 19 and 20: Section 7.2 Reducing Rational Funct
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Page 39 and 40: Section 7.3 Graphing Rational Funct
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Section 7.4 Products and Quotients
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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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