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8-2

Multiplying and Dividing Rational Expressions. 8-2. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. 1. y 2. x 6. y 5. y 3. x 2. Warm Up Simplify each expression. Assume all variables are nonzero. 1. x 5  x 2. x 7. 2. y 3  y 3. y 6. 3. 4. x 4.

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8-2

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

  2. 1 y2 x6 y5 y3 x2 Warm Up Simplify each expression. Assume all variables are nonzero. 1.x5x2 x7 2.y3y3 y6 3. 4. x4 Factor each expression. 5. x2 – 2x – 8 (x – 4)(x + 2) 6. x2 – 5x x(x – 5) 7. x5 – 9x3 x3(x – 3)(x + 3)

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

  4. Vocabulary rational expression

  5. 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

  6. 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. 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.

  7. 28x11 – 2 = 2x9 18 16x11 8x2 Check It Out! Example 1a Simplify. Identify any x-values for which the expression is undefined. Quotient of Powers Property The expression is undefined at x = 0 because this value of x makes 8x2 equal 0.

  8. 4 3 (3x + 4) 3x + 4 1 (3x + 4)(x – 1) 3x2 + x – 4 (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 0. Check It Out! Example 1b Simplify. Identify any x-values for which the expression is undefined. Factor; then divide out common factors. =

  9. 4 Check Substitute x = 1 and x = – into the original expression. 3 7 0 3(1) + 4 3(1)2 + (1) – 4 Check It Out! Example 1b Continued = Both values of x result in division by 0, which is undefined.

  10. (2x + 1)(3x + 2) 6x2 + 7x + 2 (2x + 1) (3x + 2)(2x – 3) 6x2 – 5x – 5 (2x – 3) 2 3 The expression is undefined at x =– and x = because these values of x make the factors (3x + 2) and (2x – 3) equal 0. 3 2 Check It Out! Example 1c Simplify. Identify any x-values for which the expression is undefined. Factor; then divide out common factors. =

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

  12. –1(2)(x – 5) –1(2x – 10) 10 – 2x –2 (x – 5) x – 5 x – 5 1 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. Simplify. The expression is undefined at x = 5.

  13. 10 – 2x x – 5 Check It Out! Example 2a Continued Check The calculator screens suggest that = –2 except when x = 5.

  14. –1(x)(x – 3) –1(x2– 3x) –x2 + 3x –x 1 (x – 3)(2x – 1) 2x2 – 7x + 3 2x2 – 7x + 3 2x – 1 2 The expression is undefined at x = 3 and x = . 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. Simplify.

  15. Check The calculator screens suggest that = except when x = and x = 3. –x2 + 3x –x 1 2x2 – 7x + 3 2x – 1 2 Check It Out! Example 2b Continued

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

  17. x x x7 x7      15 15 2x 2x  2 10x – 40 10(x – 4) x + 3 x + 3 2x3 20 20 (x – 2) (x – 4)(x – 2) x2 – 6x + 8 5x + 15 5(x + 3) x4 x4 3 Check It Out! Example 3 Multiply. Assume that all expressions are defined. A. B. 2 2 2 3

  18. ÷ = 1 3 1 4 2 2 4 2 3 3 You can also divide rational expressions. Recall that to divide by a fraction, you multiply by its reciprocal. 2 =

  19. 12y2 x2 x2 x2 12y2 x4y ÷   4 4 4 12y2 x4y 3y x2 Check It Out! Example 4a Divide. Assume that all expressions are defined. Rewrite as multiplication by the reciprocal. 3 1 x4 y 2

  20. ÷   4(x – 4) 4(2x – 1)(x – 3) 4(2x2 – 7x + 3) (2x + 1)(x – 4) (2x+ 1)(x – 4) 8x2 – 28x +12 2x2– 7x – 4 2x2– 7x – 4 4x2– 1 (x +3) (2x + 1)(2x – 1) (2x + 1)(2x – 1) 8x2 – 28x +12 (x + 3)(x – 3) (x + 3)(x – 3) x2 – 9 x2 – 9 4x2– 1  Check It Out! Example 4b Divide. Assume that all expressions are defined.

  21. x2 + x – 12 = –7 x + 4 = –7 (x– 3)(x + 4) (x + 4) Check It Out! Example 5a Solve. Check your solution. Note that x ≠ –4. x – 3 = –7 x = –4 Because the left side of the original equation is undefined when x = –4, there is no solution.

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

  23. 4x2 – 9 = 5 2x +3 Note that x ≠ – . = 5 (2x+ 3)(2x – 3) (2x + 3) 3 2 Check It Out! Example 5b Solve. Check your solution. 2x – 3 = 5 x = 4

  24. 4(4)2 – 9 4x2 – 9 = 5 5 2(4) + 3 2x +3 55 11 Check It Out! Example 5b Continued Check 5 5 5

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