Multiplying and Dividing Rational Expressions

Transcription

1 Multiplying and Dividing Rational Expressions

2 Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2 3. x 6 x 2 x 7 Factor each expression. 2. y 3 y 3 y 6 x 4 4. y 2 1 y 5 y 3 5. x 2 2x 8 (x 4)(x + 2) 6. x 2 5x x(x 5) 7. x 5 9x 3 x 3 (x 3)(x + 3)

3 Objectives Simplify rational expressions. Multiply and divide rational expressions.

4 rational expression Vocabulary

5 A rational expression is a quotient of two polynomials. 5 x Other examples of rational expressions include the following:

6 Because rational expressions are ratios of polynomials, you can simplify them the same way as you simplify fractions. Recall that to write a fraction in simplest form, you can divide out common factors in the numerator and denominator. Caution! When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0.

7 Example 1A: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined. 10x 8 6x x = Quotient of Powers Property 36 3 x4 The expression is undefined at x = 0 because this value of x makes 6x 4 equal 0.

8 Example 1B: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined. x 2 + x 2 x 2 + 2x 3 (x + 2)(x 1) (x 1)(x + 3) = (x + 2) (x + 3) Factor; then divide out common factors. The expression is undefined at x = 1 and x = 3 because these values of x make the factors (x 1) and (x + 3) equal 0.

9 Example 1B Continued Check Substitute x = 1 and x = 3 into the original expression. (1) 2 + (1) 2 (1) 2 + 2(1) 3 = 0 0 ( 3) 2 + ( 3) 2 ( 3) 2 + 2( 3) 3 = 4 0 Both values of x result in division by 0, which is undefined.

10 Check It Out! Example 1b Simplify. Identify any x-values for which the expression is undefined. 3x + 4 3x 2 + x 4 (3x + 4) (3x + 4)(x 1) = 1 (x 1) Factor; then divide out common factors. The expression is undefined at x = 1 and x = 4 3 because these values of x make the factors (x 1) and (3x + 4) equal 0.

11 Check It Out! Example 1b Continued Check Substitute x = 1 and x = the original expression. 4 3 into 3(1) + 4 3(1) 2 + (1) 4 = 7 0 Both values of x result in division by 0, which is undefined.

12 Check It Out! Example 1c Simplify. Identify any x-values for which the expression is undefined. 6x 2 + 7x + 2 6x 2 5x 6 (2x + 1)(3x + 2) (3x + 2)(2x 3) = (2x + 1) (2x 3) Factor; then divide out common factors. The expression is undefined at x = 2 and x = because these values of x make the factors (3x + 2) and (2x 3) equal 0.

13 Check It Out! Example 1c Continued Check Substitute x = 3 and x = 2 into the original expression. 2 3 Both values of x result in division by 0, which is undefined.

14 Example 2: Simplifying by Factoring by 1 Simplify 4x x 2. Identify any x values x 2 2x 8 for which the expression is undefined. 1(x 2 4x) x 2 2x 8 1(x)(x 4) (x 4)(x + 2) Factor out 1 in the numerator so that x 2 is positive, and reorder the terms. Factor the numerator and denominator. Divide out common factors. x (x + 2 ) Simplify. The expression is undefined at x = 2 and x = 4.

15 Example 2 Continued Check The calculator screens suggest that 4x x 2 = x except when x = 2 x 2 2x 8 (x + 2) or x = 4.

16 Check It Out! Example 2a Simplify 10 2x. Identify any x values x 5 for which the expression is undefined. 1(2x 10) x 5 1(2)(x 5) (x 5) Factor out 1 in the numerator so that x is positive, and reorder the terms. Factor the numerator and denominator. Divide out common factors. 2 1 Simplify. The expression is undefined at x = 5.

17 Check It Out! Example 2a Continued Check The calculator screens suggest that 10 2x x 5 = 2 except when x = 5.

18 Check It Out! Example 2b Simplify x 2 + 3x. Identify any x values 2x 2 7x + 3 for which the expression is undefined. 1(x 2 3x) 2x 2 7x + 3 1(x)(x 3) (x 3)(2x 1) Factor out 1 in the numerator so that x is positive, and reorder the terms. Factor the numerator and denominator. Divide out common factors. x 2x 1 Simplify. The expression is undefined at x = 3 and x =. 1 2

19 Check It Out! Example 2b Continued Check The calculator screens suggest that x 2 + 3x = x except when x = 2x 2 7x + 3 2x 1 and x =

20 You can multiply rational expressions the same way that you multiply fractions.

21 Example 3: Multiplying Rational Expressions Multiply. Assume that all expressions are defined. A. 3x 5 y 3 2x 3 y 10x3 y 4 7 9x 2 y 5 3x 5 y 3 10x3 y 4 2x 3 y 7 9x 2 y 5 5x 3 3y B. x 3 4x + 20 x + 5 x 2 9 x 3 4(x + 5) x + 5 (x 3)(x + 3) 1 4(x + 3)

22 Check It Out! Example 3 Multiply. Assume that all expressions are defined. A. 3 x 15 x 15 x x 2x x 7 2 x 4 20 x 4 2 B. 10x 40 x 2 6x + 8 x + 3 5x (x 4) (x 4)(x 2) x + 3 5(x + 3) 2x (x 2)

23 You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal = = 2 3

24 Example 4A: Dividing Rational Expressions Divide. Assume that all expressions are defined. 5x 4 8x 2 y y 5 5x 4 8x 2 y 8y Rewrite as multiplication by the reciprocal. 5x 4 8x 2 y 8y x 2 y

25 Example 4B: Dividing Rational Expressions Divide. Assume that all expressions are defined. x 4 9x 2 x 2 4x + 3 x4 + 2x 3 8x 2 x 2 16 x 4 9x 2 x 2 4x + 3 x 2 16 x 4 + 2x 3 8x 2 x 2 (x 2 9) x 2 4x + 3 x 2 16 x 2 (x 2 + 2x 8) x 2 (x 3)(x + 3) (x 3)(x 1) (x + 3)(x 4) (x 1)(x 2) (x + 4)(x 4) x 2 (x 2)(x + 4) Rewrite as multiplication by the reciprocal.

26 2x 2 7x 4 x 2 9 Check It Out! Example 4b Divide. Assume that all expressions are defined. 2x 2 7x 4 x 2 9 4x 2 1 8x 2 28x +12 8x 2 28x +12 4x 2 1 (2x + 1)(x 4) (x + 3)(x 3) 4(2x2 7x + 3) (2x + 1)(2x 1) (2x + 1)(x 4) (x + 3)(x 3) 4(x 4) (x +3) 4(2x 1)(x 3) (2x + 1)(2x 1)

27 Example 5A: Solving Simple Rational Equations Solve. Check your solution. x 2 25 x 5 = 14 (x + 5)(x 5) = 14 (x 5) x + 5 = 14 x = 9 Note that x 5.

28 Example 5A Continued Check x 2 25 x 5 = 14 (9)

29 Check It Out! Example 5a Solve. Check your solution. x 2 + x 12 x + 4 = 7 (x 3)(x + 4) = 7 (x + 4) x 3 = 7 Note that x 4. x = 4 Because the left side of the original equation is undefined when x = 4, there is no solution.

30 Check It Out! Example 5a Continued Check A graphing calculator shows that 4 is not a solution.

31 Check It Out! Example 5b Solve. Check your solution. 4x 2 9 2x + 3 = 5 (2x + 3)(2x 3) = 5 (2x + 3) 2x 3 = 5 x = 4 Note that x. 3 2

32 Check It Out! Example 5b Continued Check 4x 2 9 2x + 3 = 5 4(4) 2 9 2(4)