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Holt McDougal Algebra Multiplying and Dividing Rational Expressions 8-2 Multiplying and Dividing Rational Expressions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

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8-2 Multiplying and Dividing Rational Expressions Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5 x 2 3. x7x7 x 6 x2x2 Factor each expression. 5. x 2 – 2x – 8 6. x 2 – 5x (x – 4)(x + 2) x 3 (x – 3)(x + 3) 2. y 3 y 3 y6y6 x4x4 y 2 y5y y3y3 7. x 5 – 9x 3 x(x – 5)

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Simplify rational expressions. Multiply and divide rational expressions. Objectives

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions rational expression Vocabulary

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions In Lesson 8-1, you worked with inverse variation functions such as y =. The expression on the right side of this equation is a rational expression. A rational expression is a quotient of two polynomials. Other examples of rational expressions include the following: 5 x

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions When identifying values for which a rational expression is undefined, identify the values of the variable that make the original denominator equal to 0. Caution! 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.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Simplify. Identify any x-values for which the expression is undefined. Example 1A: Simplifying Rational Expressions Quotient of Powers Property 10x 8 6x46x4 5 10x 8 – x4x4 = The expression is undefined at x = 0 because this value of x makes 6x 4 equal 0.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions 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) Factor; then divide out common factors. = (x + 2) (x + 3) 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.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check Substitute x = 1 and x = –3 into the original expression. (1) 2 + (1) – 2 (1) 2 + 2(1) – = (–3) 2 + (–3) – 2 (–3) 2 + 2(–3) – = Both values of x result in division by 0, which is undefined. Example 1B Continued

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 1a Simplify. Identify any x-values for which the expression is undefined. Quotient of Powers Property 16x 11 8x28x2 2 8x 11 – x92x9 = The expression is undefined at x = 0 because this value of x makes 8x 2 equal 0.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions 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) Factor; then divide out common factors. = 1 (x – 1) The expression is undefined at x = 1 and x = – because these values of x make the factors (x – 1) and (3x + 4) equal

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions 3(1) + 4 3(1) 2 + (1) – = Both values of x result in division by 0, which is undefined. Check It Out! Example 1b Continued Check Substitute x = 1 and x = – into the original expression. 4 3

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

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Both values of x result in division by 0, which is undefined. Check Substitute x = and x = – into the original expression Check It Out! Example 1c Continued

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Simplify. Identify any x values for which the expression is undefined. Example 2: Simplifying by Factoring by –1 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. The expression is undefined at x = –2 and x = 4. 4x – x 2 x 2 – 2x – 8 –1(x 2 – 4x) x 2 – 2x – 8 –1(x)(x – 4) (x – 4)(x + 2) –x–x (x + 2 ) Simplify.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check The calculator screens suggest that = except when x = – 2 or x = 4. Example 2 Continued 4x – x 2 x 2 – 2x – 8 –x–x (x + 2)

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

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 2a Continued Check The calculator screens suggest that = –2 except when x = – 2x x – 5

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 2b Simplify. Identify any x values for which the expression is undefined. 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 2 + 3x 2x 2 – 7x + 3 –1(x 2 – 3x) 2x 2 – 7x + 3 –x–x 2x – 1 Simplify. –1(x)(x – 3) (x – 3)(2x – 1) The expression is undefined at x = 3 and x =. 1 2

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 2b Continued –x 2 + 3x 2x 2 – 7x + 3 Check The calculator screens suggest that = except when x = and x = 3. –x–x 2x – 1 1 2

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions You can multiply rational expressions the same way that you multiply fractions.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Multiply. Assume that all expressions are defined. Example 3: Multiplying Rational Expressions A. 3x 5 y 3 2x3y72x3y7 10x 3 y 4 9x2y59x2y5 3x 5 y 3 2x3y72x3y7 10x 3 y 4 9x2y59x2y x 3 3y 5 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)

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 3 Multiply. Assume that all expressions are defined. A. x 15 20 x4x4 2x2x x 7 x 15 20 x4x4 2x2x x7 x x 3 3 B. 10x – 40 x 2 – 6x + 8 x + 3 5x (x – 4) (x – 4)(x – 2) x + 3 5(x + 3) 2 (x – 2) 2

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal ÷ = =

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Divide. Assume that all expressions are defined. Example 4A: Dividing Rational Expressions Rewrite as multiplication by the reciprocal. 5x 4 8x2y28x2y2 ÷ 8y58y5 15 5x 4 8x2y28x2y2 15 8y58y5 5x4 5x4 8x2y28x2y2 8y58y x 2 y 3 3

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Example 4B: Dividing Rational Expressions x 4 – 9x 2 x 2 – 4x + 3 ÷ x 4 + 2x 3 – 8x 2 x 2 – 16 Divide. Assume that all expressions are defined. x 4 – 9x 2 x 2 – 4x + 3 x 2 – 16 x 4 + 2x 3 – 8x 2 Rewrite as multiplication by the reciprocal. 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 + 4)(x – 4) x 2 (x – 2)(x + 4) (x + 3)(x – 4) (x – 1)(x – 2)

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions x 4 y Check It Out! Example 4a Rewrite as multiplication by the reciprocal. x 2 4 ÷ 12y 2 x4yx4y x2 x2 4 2 3y x2 x2 Divide. Assume that all expressions are defined. x 2 4 x 4 y 12y 2 31

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 4b 2x 2 – 7x – 4 x 2 – 9 ÷ 4x 2 – 1 8x 2 – 28x +12 Divide. Assume that all expressions are defined. (2x + 1)(x – 4) (x + 3)(x – 3) 4(2x 2 – 7x + 3) (2x + 1)(2x – 1) (2x + 1)(x – 4) (x + 3)(x – 3) 4(2x – 1)(x – 3) (2x + 1)(2x – 1) 4(x – 4) (x +3) 2x 2 – 7x – 4 x 2 – 9 8x 2 – 28x +12 4x 2 – 1

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Example 5A: Solving Simple Rational Equations Solve. Check your solution. Note that x ≠ 5. x 2 – 25 x – 5 = 14 (x + 5)(x – 5) (x – 5) = 14 x + 5 = 14 x = 9

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions x 2 – 25 x – 5 = 14 Check (9) 2 – 25 9 – Example 5A Continued

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Example 5B: Solving Simple Rational Equations Solve. Check your solution. Note that x ≠ 2. x 2 – 3x – 10 x – 2 = 7 (x + 5)(x – 2) (x – 2) = 7 x + 5 = 7 x = 2 Because the left side of the original equation is undefined when x = 2, there is no solution.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check A graphing calculator shows that 2 is not a solution. Example 5B Continued

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 5a Solve. Check your solution. Note that x ≠ –4. x 2 + x – 12 x + 4 = –7 (x – 3)(x + 4) (x + 4) = –7 x – 3 = –7 x = –4 Because the left side of the original equation is undefined when x = –4, there is no solution.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Check It Out! Example 5a Continued Check A graphing calculator shows that –4 is not a solution.

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Solve. Check your solution. 4x 2 – 9 2x + 3 = 5 (2x + 3)(2x – 3) (2x + 3) = 5 2x – 3 = 5 x = 4 Check It Out! Example 5b Note that x ≠ –. 3 2

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions 4x 2 – 9 2x + 3 = 5 Check 4(4) 2 – 9 2(4) Check It Out! Example 5b Continued

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Lesson Quiz: Part I Simplify. Identify any x-values for which the expression is undefined. x 2 – 6x + 5 x 2 – 3x – 10 6x – x 2 x 2 – 7x + 6 x – 1 x + 2 x ≠ –2, 5 –x–x x – 1 x ≠ 1, 6

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Holt McDougal Algebra Multiplying and Dividing Rational Expressions Lesson Quiz: Part II 3. Multiply or divide. Assume that all expressions are defined. x 2 + 4x + 3 x 2 – 4 ÷ x 2 + 2x – 3 x 2 – 6x x + 1 3x + 6 6x + 12 x 2 – 1 (x + 1)(x – 4) (x + 2)(x – 1) 2 x – 1 5. x = 4 Solve. Check your solution. 4x 2 – 1 2x – 1 = 9

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