# Multiplying and Dividing Rational Expressions

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

Simplify each expression. Assume all variables are nonzero.
Warm Up Simplify each expression. Assume all variables are nonzero. 1. x5  x2 x7 2. y3  y3 y6 3. x6 x2 4. y2 y5 1 y3 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)

Objectives Simplify rational expressions.
Multiply and divide rational expressions.

Vocabulary rational expression

In Lesson 8-1, you worked with inverse variation functions such as y =
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

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. 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!

Example 1A: Simplifying Rational Expressions
Simplify. Identify any x-values for which the expression is undefined. 10x8 6x4 510x8 – 4 36 5 3 x4 = Quotient of Powers Property The expression is undefined at x = 0 because this value of x makes 6x4 equal 0.

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 – 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.

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

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

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 = –because these values of x make the factors (x – 1) and (3x + 4) equal 0. 4 3

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

Check It Out! Example 1c Simplify. Identify any x-values for which the expression is undefined. 6x2 + 7x + 2 6x2 – 5x – 5 (2x + 1)(3x + 2) (3x + 2)(2x – 3) (2x + 1) (2x – 3) Factor; then divide out common factors. = 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 Continued
Check Substitute x = and x = – into the original expression. 3 2 Both values of x result in division by 0, which is undefined.

Example 2: Simplifying by Factoring by –1
Simplify Identify any x values for which the expression is undefined. 4x – x2 x2 – 2x – 8 –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.

Example 2 Continued Check The calculator screens suggest that = except when x = – or x = 4. 4x – x2 x2 – 2x – 8 –x (x + 2)

Check It Out! Example 2a Simplify Identify any x values for which the expression is undefined. 10 – 2x x – 5 –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.

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

Check It Out! Example 2b Simplify Identify any x values for which the expression is undefined. –x2 + 3x 2x2 – 7x + 3 –1(x2 – 3x) 2x2 – 7x + 3 Factor out –1 in the numerator so that x is positive, and reorder the terms. –1(x)(x – 3) (x – 3)(2x – 1) Factor the numerator and denominator. Divide out common factors. –x 2x – 1 Simplify. The expression is undefined at x = 3 and x = . 1 2

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

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

Example 3: Multiplying Rational Expressions
Multiply. Assume that all expressions are defined. 3x5y3 2x3y7 10x3y4 9x2y5 x – 3 4x + 20 x + 5 x2 – 9 A. B. 3x5y3 2x3y7 10x3y4 9x2y5 3 5 x – 3 4(x + 5) x + 5 (x – 3)(x + 3) 3 5x3 3y5 1 4(x + 3)

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

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

Example 4A: Dividing Rational Expressions
Divide. Assume that all expressions are defined. 5x4 8x2y2 ÷ 8y5 15 5x4 8x2y2 15 8y5 Rewrite as multiplication by the reciprocal. 5x4 8x2y2 15 8y5 2 3 3 x2y3 3

Example 4B: Dividing Rational Expressions
Divide. Assume that all expressions are defined. x4 – 9x2 x2 – 4x + 3 ÷ x4 + 2x3 – 8x2 x2 – 16 Rewrite as multiplication by the reciprocal. x4 – 9x2 x2 – 4x + 3 x2 – 16 x4 + 2x3 – 8x2 x2 (x2 – 9) x2 – 4x + 3 x2 – 16 x2(x2 + 2x – 8) x2(x – 3)(x + 3) (x – 3)(x – 1) (x + 4)(x – 4) x2(x – 2)(x + 4) (x + 3)(x – 4) (x – 1)(x – 2)

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

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

Example 5A: Solving Simple Rational Equations
Solve. Check your solution. x2 – 25 x – 5 = 14 (x + 5)(x – 5) (x – 5) = 14 Note that x ≠ 5. x + 5 = 14 x = 9

Example 5A Continued x2 – 25 x – 5 = 14 Check (9)2 – 25 9 – 5 14 56 4 14

Example 5B: Solving Simple Rational Equations
Solve. Check your solution. x2 – 3x – 10 x – 2 = 7 (x + 5)(x – 2) (x – 2) = 7 Note that x ≠ 2. x + 5 = 7 x = 2 Because the left side of the original equation is undefined when x = 2, there is no solution.

Example 5B Continued Check A graphing calculator shows that 2 is not a solution.

Check It Out! Example 5a Solve. Check your solution. x2 + x – 12 x + 4 = –7 (x – 3)(x + 4) (x + 4) = –7 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.

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

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

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

Lesson Quiz: Part I Simplify. Identify any x-values for which the expression is undefined. 1. x2 – 6x + 5 x2 – 3x – 10 x – 1 x + 2 x ≠ –2, 5 6x – x2 x2 – 7x + 6 –x x – 1 2. x ≠ 1, 6

Lesson Quiz: Part II Multiply or divide. Assume that all expressions are defined. 3. x + 1 3x + 6 6x + 12 x2 – 1 2 x – 1 4. x2 + 4x + 3 x2 – 4 ÷ x2 + 2x – 3 x2 – 6x + 8 (x + 1)(x – 4) (x + 2)(x – 1) Solve. Check your solution. 4x2 – 1 2x – 1 = 9 5. x = 4