8.2 M ULTIPLYING AND D IVIDING R ATIONAL E XPRESSIONS Simplify rational expressions. Multiply and divide rational expressions. Objectives rational expression Vocabulary Why Learning This? You can simplify rational expressions to determine the probability of hitting an archery target. Warm Up Simplify each expression. 1. x 5 x 2 Factor each expression. 5. x 2 – 2 x – 8 6. x 2 – 5 x 2. y 2 y5y5 7. x 5 – 9 x 3
A rational expression is a quotient of two polynomials. Simplifying fractions: Simplify. Identify any x -values for which the expression is undefined. Example 1A: Simplifying Rational Expressions 10 x 8 6x46x4 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! x 2 + x – 2 x x – 3 3 x x 2 + x – 4 16 x 11 8x28x2
Simplify. Identify x values for which the expression is undefined. Example 2: Simplifying by Factoring by –1 4 x – x 2 x 2 – 2 x – 8 Check The calculator screens suggest that = except when x = – 2 or x = 4. 4 x – x 2 x 2 – 2 x – 8 –x–x ( x + 2) – x x 2 x 2 – 7 x + 3 A.B.
Multiply. Assume that all expressions are defined. Example 3: Multiplying Rational Expressions A. 3 x 5 y 3 2x3y72x3y7 10 x 3 y 4 9x2y59x2y5 B. x – 3 4 x + 20 x + 5 x 2 – 9
Divide. Assume that all expressions are defined. Example 4A: Dividing Rational Expressions 5 x 4 8x2y28x2y2 ÷ 8y58y5 15 Recall that to divide by a fraction, you multiply by its reciprocal ÷ = = x 4 – 9 x 2 x 2 – 4 x + 3 ÷ x x 3 – 8 x 2 x 2 – 16
Example 5A: Solving Simple Rational Equations Solve. Check your solution. Note that x ≠ 5. x 2 – 25 x – 5 = 14 x 2 + x – 12 x + 4 = 0 x 2 + x – 12 = 0 What are the solutions? Will we have the same solutions below? Factor Divide out Common Factors Check:
Example 5B: Solving Simple Rational Equations Solve. Check your solution. Note that x ≠ 2. x 2 – 3 x – 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. Check