Multiplication and Division of Rational Expressions and Functions

Transcription

1 7.2 Multiplication and Division of Rational Expressions and Functions 7.2 OBJECTIVES 1. Multiply two rational expressions 2. Divide two rational expressions 3. Multiply two rational functions 4. Divide two rational functions Once again, let s turn to an example from arithmetic to begin our discussion of multiplying rational expressions. Recall that to multiply two fractions, we multiply the numerators and multiply the denominators. For instance, In algebra, the pattern is exactly the same. Rules and Properties: Multiplying Rational Expressions For polynomials P, Q, R, and S, NOTE For all problems with rational expressions, assume denominators are not 0. P Q R PR S QS when Q 0 and S 0 Example 1 Multiplying Rational Expressions Multiply. 2x 3 5y 2 10y 3x 2 20x3 y 15x 2 y 2 5x2 y 4x 5x 2 y 3y Divide by the common factor 5x 2 y to simplify. 4x 3y CHECK YOURSELF 1 NOTE The factoring methods in Chapter 6 are used to simplify rational expressions. Multiply. 9a 2 b 3 5ab 4 20ab2 27ab 3 Generally, you will find it best to divide by any common factors before you multiply, as Example 2 illustrates. 499

2 500 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS Example 2 Multiplying Rational Expressions Multiply as indicated. (a) x 6x 18 x 2 3x 9x 1 2 x x(x 3) 6(x 3) 9x Factor. Divide by the common factors of 3, x, and x x x 2 y 2 5x 2 5xy 10xy x 2 2xy y 2 Factor and divide by the common factors of 5, x, x y, and x y (x y)(x y) 10xy 5x(x y) (x y)(x y) y x y NOTE From Section 7.1, recall that 2 x x x 5x2 x 2 2x 8x x(x 2) 5x(2 x) 8(x 3) 2 5 2(x 3) CHECK YOURSELF 2 Multiply as indicated. (a) x 2 5x 14 8x 56 4x 2 x 2 49 x 3x x2 2x 6 2 The following algorithm summarizes our work in multiplying rational expressions. Step by Step: Step 1 Step 2 Step 3 Multiplying Rational Expressions Write each numerator and denominator in completely factored form. Divide by any common factors appearing in both the numerator and denominator. Multiply as needed to form the product.

3 MULTIPLICATION AND DIVISION OF RATIONAL EXPRESSIONS AND FUNCTIONS SECTION In dividing rational expressions, you can again use your experience from arithmetic. Recall that NOTE We invert the divisor (the second fraction) and multiply Once more, the pattern in algebra is identical. Rules and Properties: Dividing Rational Expressions For polynomials P, Q, R, and S, P Q R S P Q S PS R QR when Q 0 R 0 and S 0 To divide rational expressions, invert the divisor and multiply as before, as Example 3 illustrates. Example 3 Dividing Rational Expressions Divide as indicated. NOTE Invert the divisor and multiply. (a) 3x 2 8x 3 y 9x2 y 2 4y 4 3x 2 8x 3 y 4y 4 9x 2 y 2 y 6x 3 CAUTION Be Careful! Invert the divisor, then factor. 2x 2 4xy 9x 18y 4x 8y 3x 6y 2x 2 4xy 9x 18y 3x 6y 4x 8y 1 1 2x(x 2y) 3(x 2y) 9(x 2y) 4(x 2y) x x 2 x 6 4x 2 6x x2 4 4x 2x 2 x 6 4x 2 6x 4x x (2x 3) (x 2) 4x 2x (2x 3) (x 2)(x 2) 2 x CHECK YOURSELF 3 Divide and simplify. (a) 5xy 10y2 3 7x 14x 3 x 2 9 x 3 27 x2 2x 15 2x 2 10x 3x 9y 2x 10y x2 3xy 4x 2 20xy

4 502 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS We summarize our work in dividing fractions with the following algorithm. Step by Step: Dividing Rational Expressions Step 1 Step 2 Invert the divisor (the second rational expression) to write the problem as one of multiplication. Proceed as in the algorithm for the multiplication of rational expressions. The product of two rational functions is always a rational function. Given two rational functions, f(x) and g(x), we can rename the product, so h(x) f(x) g(x) This will always be true for values of x for which both f and g are defined. So, for example, h(1) f(1) g(1) as long as both f(1) and g(1) exist. Example 4 illustrates this concept. Example 4 Multiplying Rational Functions Given the rational functions f(x) x2 3x 10 x 1 find the following. and g(x) x2 4x 5 x 5 (a) f(0) g(0) Because f(0) 10 and g(0) 1, then f(0) g(0) (10)(1) 10. f(5) g(5) Although we can find f(5), g(5) is undefined. 5 is an excluded value for the domain of the function g. Therefore, f(5) g(5) is undefined. h(x) f(x) g(x) h(x) f(x) g(x) x2 3x 10 x2 4x 5 x 1 x (x 5)(x 2) (x 1)(x 5) (x 1) (x 5) 1 1 (x 5)(x 2) x 1, x 5 h(0) h(0) (0 5)(0 2) 10 (e) h(5) Although the temptation is to substitute 5 for x in part, notice that the function is undefined when x is 1 or 5. As was true in part, the function is undefined at that point.

5 MULTIPLICATION AND DIVISION OF RATIONAL EXPRESSIONS AND FUNCTIONS SECTION CHECK YOURSELF 4 Given the rational functions f(x) x2 2x 8 x 2 and g(x) x2 3x 10 x 4 find the following. (a) f(0) g(0) f(4) g(4) h(x) f(x) g(x) h(0) (e) h(4) When we divide two rational functions to create a third rational function, we must be certain to exclude values for which the denominator is equal to zero, as Example 5 illustrates. Example 5 Dividing Polynomial Functions Given the rational functions f(x) x3 2x 2 x 2 complete the following. f(0) (a) Find g(0). 1 Because f(0) 0 and g(0), then 2 f(0) g(0) and g(x) x2 3x 2 x 4 f(1) Find g(1). Although we can find both f(1) and g(1), g(1) 0, so division is undefined when x 1. 1 is an excluded value for the domain of the quotient. Find h(x) h(x) f(x) g(x) x 3 2x 2 x 2 x 2 3x 2 x 4 x3 2x 2 x 2 x 4 1 x2 (x 2) x 2 f(x) g(x). x2 (x 4) (x 2)(x 1) Invert and multiply. x 2 3x 2 x 4 (x 1)(x 2) 1 x 2, 1, 2, 4

6 504 CHAPTER 7 RATIONAL EXPRESSIONS AND FUNCTIONS For which values of x is h(x) undefined? h(x) will be undefined for any value of x for which f(x) is undefined, g(x) is undefined, or g(x) 0. h(x) is undefined for the values 2, 1, 2, and 4. CHECK YOURSELF 5 Given the rational functions f(x) x2 2x 1 x 3 and g(x) x2 5x 4 x 2 complete the following. f(0) f(1) f(x) (a) Find Find Find h(x). g(0). g(1) g(x) For which values of x is h(x) undefined? CHECK YOURSELF ANSWERS 4a 2(x 2) x 2 x 2x (a) 3. (a) 6; x 2 ; 3b 2 4 y ; x 2 3x 9 4. (a) 10; undefined; h(x) (x 5)(x 2) x 4, x 2; 10; and (e) undefined 5. (a) 1 undefined; h(x) 6 ; x 3, 1, 2, 4 (x 1)(x 2) (x 3)(x 4) ; and

7 Name 7.2 Exercises Section Date In exercises 1 to 36, multiply or divide as indicated. Express your result in simplest form. x 2 6x x 4 y y y 6 ANSWERS a a a 21 p 5 8 p2 12p xy 2 25xy x 16y 3 3x 3 y 5xy2 3 10xy 9xy b 3 2ab ab 20ab 3 4x 2 y 2 9x 3 8y2 27xy m 3 n 6mn2 3mn mn m 3 n 5m 2 n 4cd 2 5cd 3c3 d 2c 2 d 9cd 20cd x 15 9x x 2x 6 a 2 3a 5a 20a 2 3a b 15 4b b 9b 2 7m 2 28m 4m 5m 20 12m x 2 3x 10 15x x 3x 15 c 2 2c 8 5c c 18c y 2 8y 4y m m 12y 2 y m 21 20m

8 ANSWERS 19. x 2 2x 8 10x x 16 x 2 4 y 2 7y 10 y 2 5y 2y y d 2 3d d d 96 20d b 2 6b 8 b 2 4b b2 4 2b x 2 x x 2 7x 4 3x2 11x 20 4x 2 9 4p 2 1 2p 2 9p 5 3p2 13p 10 9p a a 2 6a 2a2 5a 3 4a 2 1 2x 2 5x 7 4x 2 9 5x2 5x 2x 2 3x w w 2 2w 3w 3 w 3y 15 y 2 3y 4y 5 y a a a 6 a 2 3a x x x 2 2x 3x x 2 9y 2 4x 10y x 2 2 xy 15y x 2 3xy 2a 2 7ab 15b 2 2ab 10b 2 2a2 3ab 4a 2 9b m 2 5mn 2n m 2 4n 2 m3 m 2 n 9m 2 6mn 2x 2 y 5xy 2 4x 2 25y 2 4x 2 20xy 2x 2 15xy 25y (a) (e) x x 2 4 5x 10 x 3 2x 2 4x a 3 27 a 2 9 a3 3a 2 9a 3a 3 9a Let f(x) x2 3x 4 and g(x) x2 2x 8. Find (a) f(0) g(0), x 2 x 4 f(4) g(4), h(x) f(x) g(x), h(0), and (e) h(4). 506

9 ANSWERS 38. Let f(x) x2 4x 3 and g(x) x2 7x 10. Find (a) f(1) g(1), x 5 x 3 f(3) g(3), h(x) f(x) g(x), h(1), and (e) h(3). 39. Let f(x) 2x2 3x 5 and g(x) 3x2 5x 2. Find (a) f(1) g(1), x 2 x 1 f(2) g(2), h(x) f(x) g(x), h(1), and (e) h(2). 40. Let f(x) x2 1 Find (a) f(2) g(2), f(3) g(3), x 3 and g(x) x2 9 x 1. h(x) f(x) g(x), h(2), and (e) h(3). 38. (a) (e) 39. (a) (e) 40. (a) (e) 41. (a) f(0) f(1) 41. Let f(x) 3x2 x 2 and g(x) x2 4x 5. Find (a),, x 2 x 4 g(0) g(1) h(x) f(x) the values of x for which h(x) is undefined. g(x), and 42. (a) f(0) f(2) 42. Let f(x) x2 x Find (a),, x 5 and g(x) x2 x 6. x 5 g(0) g(2) h(x) f(x) the values of x for which h(x) is undefined. g(x), and The results from multiplying and dividing rational expressions can be checked by using a graphing calculator. To do this, define one expression in Y 1 and the other in Y 2. Then define the operation in Y 3 as Y 1 Y 2 or Y 1 Y 2. Put your simplified result in Y 4 (sorry, you still must simplify algebraically). Deselect the graphs for Y 1 and Y 2. If you have correctly simplified the expression, the graphs of Y 3 and Y 4 will be identical. Use this technique in exercises 43 to x 3 3x 2 2x x 2 5x2 15x 9 20x x x 2 x 6 (x3 4x) 3a 3 a 2 9a 3 15a 2 5a 3a2 9 a 4 9 w 3 27 w 2 2w 3 (w3 3w 2 9w) 507

10 Answers b 3 15x 9b mn x a 4 12x 3a x 2 3(c 2) 5x 5d x 5 2x (x 2) 4(d 3) 2x 3 2a 1 6 a a w 2 6 x m x 37. (a) 4; undefined; (x 1)(x 4) x 2, 4; 4; (e) undefined 39. (a) 6; undefined; (2x 5)(3x 1) x 2, 1; 6; (e) undefined 5 (3x 2)(x 4) 41. (a) 4 x 4, 1, 2, 5; 2, 4, 1, 5 5 ; 4 ; (x 2)(x 5) x 2 2 x x(x 3) 508