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Multiplying and Dividing Rational Expressions 8-2 A rational expression is a quotient of two polynomials. 5 x A rational expression is undefined where the denominator is equal to zero. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Example 1A: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined. 10x8 6x4 510x8 – 4 5 x4 Quotient of Powers Property = 3 36 The expression is undefined at x = 0 because this value of x makes 6x4 equal 0. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Example 1B: Simplifying Rational Expressions Simplify. Identify any x-values for which the expression is undefined. x2 + x – 2 x2 + 2x – 3 (x + 2)(x – 1) = (x + 2) (x – 1)(x + 3) (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. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Example 1B Continued Check Substitute x = 1 and x = –3 into the original expression. (1)2 + (1) – 2 0 = (1)2 + 2(1) – 3 0 (–3)2 + (–3) – 2 4 = (–3)2 + 2(–3) – 3 0 Both values of x result in division by 0, which is undefined. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Check It Out! Example 1b Simplify. Identify any x-values for which the expression is undefined. 3x + 4 3x2 + 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. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Check It Out! Example 1b Continued Check Substitute x = 1 and x = – 4 3 into the original expression. 3(1) + 4 7 = 3(1)2 + (1) – 4 0 Both values of x result in division by 0, which is undefined. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Example 2: Simplifying by Factoring by –1 2 4x – x Simplify . Identify any x values 2 x – 2x – 8 for which the expression is undefined. –1(x2 – 4x) x2 – 2x – 8 Factor out –1 in the numerator so that x2 is positive, and reorder the terms. –1(x)(x – 4) (x – 4)(x + 2) Factor the numerator and denominator. Divide out common factors. –x (x + 2 ) Simplify. The expression is undefined at x = –2 and x = 4. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Check It Out! Example 2a 10 – 2x . Identify any x values x–5 for which the expression is undefined. Simplify –1(2x – 10) x–5 Factor out –1 in the numerator so that x is positive, and reorder the terms. –1(2)(x – 5) (x – 5) Factor the numerator and denominator. Divide out common factors. –2 1 Simplify. The expression is undefined at x = 5. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions You can multiply rational expressions the same way that you multiply fractions. Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Example 3: Multiplying Rational Expressions Multiply. Assume that all expressions are defined. 5y3 3y4 3x 10x A. 3 7 2x y 9x2y5 3 5 3x y3 2x3y7 5x3 3y5 Holt Algebra 2 5 3y4 10x 2 5 3 9x y B. x–3 x+5 4x + 20 x2 – 9 x–3 x+5 4(x + 5) (x – 3)(x + 3) 1 4(x + 3) 8-2 Multiplying and Dividing Rational Expressions Check It Out! Example 3 Multiply. Assume that all expressions are defined. C. 2 10x – 40 x + 3 x2 – 6x + 8 5x + 15 10(x – 4) (x – 4)(x – 2) 2 (x – 2) Holt Algebra 2 x+3 5(x + 3) 8-2 Multiplying and Dividing Rational Expressions To divide rational expressions, multiply by the reciprocal of the dividing expression. 5x4 15 ÷ 8x2y2 8y5 5x4 8y5 2 2 8x y 15 5x4 8y5 2 2 8x y 15 x2y3 3 Holt Algebra 2 Rewrite as multiplication by the reciprocal. 8-2 Multiplying and Dividing Rational Expressions Example 4B: Dividing Rational Expressions Divide. Assume that all expressions are defined. 4 + 2x3 – 8x2 x4 – 9x2 x ÷ 2 x – 4x + 3 x2 – 16 x4 – 9x2 2 x – 4x + 3 x2 – 16 x4 + 2x3 – 8x2 Rewrite as multiplication by the reciprocal. x2 (x2 – 9) x2 – 16 x2 – 4x + 3 x2(x2 + 2x – 8) x2(x – 3)(x + 3) (x + 4)(x – 4) (x – 3)(x – 1) x2(x – 2)(x + 4) (x + 3)(x – 4) (x – 1)(x – 2) Holt Algebra 2 8-2 Multiplying and Dividing Rational Expressions Check It Out! Example 4b Divide. Assume that all expressions are defined. 2x2 – 7x – 4 ÷ 4x2– 1 x2 – 9 8x2 – 28x +12 2x2 – 7x – 4 x2 – 9 8x2 – 28x +12 4x2– 1 (2x + 1)(x – 4) 4(2x2 – 7x + 3) (x + 3)(x – 3) (2x + 1)(2x – 1) (2x + 1)(x – 4) 4(2x – 1)(x – 3) (x + 3)(x – 3) (2x + 1)(2x – 1) 4(x – 4) (x +3) Holt Algebra 2 Multiplying and Dividing Rational Expressions 8-2 Check It Out! Example 5a Solve. x2 + x – 12 = –7 x+4 (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. Holt Algebra 2 Multiplying and Dividing Rational Expressions 8-2 Check It Out! Example 5b Solve. 4x2 – 9 =5 2x + 3 (2x + 3)(2x – 3) = 5 (2x + 3) 2x – 3 = 5 x=4 Holt Algebra 2 Note that x ≠ – 3 . 2 8-2 Multiplying and Dividing Rational Expressions HW pg. 580 #’s 19 – 31 HW pg. 580 #’s 15-17, 32-34, 42, 43, 45,46 Holt Algebra 2