8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression: Factor the numerator and the denominator completely. Divide out any common factors. Identify any x-values for which the expression is undefined. Simplify: 24 x 6 8 x 2. 24 x 6 8 x 8 $ 3 2 8 x 6 2 3 x 4 x 0, because 8 x 2 is undefined at x 0. Simplify: x _ 2 2x 8 x 2 x 2. First, factor the numerator and the denominator. Use the Quotient of Powers Property. x 2 2x 8 x 2 x 2 x 4 x 2 x 2 x 1 x 4 x 2 x 2 x 1 x 4 x 1 x 4 x 1 x 2 and x 1 Divide out common factors. Simplify. 1. x 2 2x 3 x 2 6x 5 x 1 x 3 x 1 x 5 2. 20 x 9 4 x 3. x 2 4x 3 x 2 5x 4 20 x x 4 4 x 9 x 3 x 4 x 1 x 3 x 5, x 5 x 6 x 1 x 1, 5 x 0 x 1, 4 14 Holt Algebra 2
8-2 Multiplying and Dividing Rational Expressions (continued) Multiplying rational expressions is similar to multiplying fractions. Multiply: _5 x 2 y 3 4 x 3 y 2 x 4 y 3. 5 3x y 2 _5 x 2 y 3 4 x 3 y 2 x 4 y 3 5 3x y 15 2 4 2 3 x 2 x 4 x 3 x y 3 y 3 y 5 y 2 5 2 x 6 x y 6 4 y 7 5 2 x 2 1 y 5 x 2 2y Simplify. Group like factors. Simplify constants. Add exponents to multiply. Subtract exponents to divide. Multiplying rational expressions is similar to simplifying rational expressions. Multiply: x 3 _ 6x 6 x 3 _ 6x 6 _ x 1 x 2 9. _ x 1 x 2 9 x 3 6 x 1 x 1 x 3 x 3 x 3 6 x 1 x 1 x 3 x 3 1 6 x 3 To divide rational expressions, multiply by the reciprocal. x 7 x 2 x 2 49 2x 4 x 7 x 2 Simplify. 2x 4 x 2 49 x 7 x 2 Completely factor all numerators and denominators. Divide out common factors. 2 x 2 x 7 x 7 2 x 7 Multiply. Assume that all expressions are defined. 4. _2 x 5 y 2 6 x 2 y 9 x 3 y 5. 4 3 x 2 y 3 6 x 4 y 4 7. 3 x 3 y 9 xy 3 5x y 2 15y x 8. 2x _ 2 x 4 x 2 4x x 2 3x 2 2x x 2 4x _ 8 x 2 4 3x x 2 4 y 4 3x 6. 8x 16 x 2 1 9. x 2 2x 3 x 2 9 x 1 _ 4x 8 2 x 1 x 2 3x 4 x 2 2x 3 x 1 x 4 15 Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions Use a common denominator to add or subtract rational expressions. Add: 6x _ 4 2x _ 8 x 5 x 5. Step 1 Add. 6x 4 _ x 5 2x _ 8 x 5 6x 4 2x 8 x 5 6x 2x 4 8 x 5 8x _ 4 x 5 The denominators are the same. Add the numerators. Group like terms. Combine like terms. Step 2 Identify x-values for which the expression is undefined. Subtract: x 5 because 5 makes the denominator equal 0. 4x _ 3 2x 1 8x 2 _ 2x 1. Step 1 Subtract. 4x _ 3 8x _ 2 2x 1 2x 1 4x 3 8x 2 2x 1 4x 3 8x 2 2x 1 4x 5 2x 1 Step 2 Identify x-values for which the expression is undefined. x 1 because 1 makes the denominator equal 0. 2 2 Add or subtract. 1. _ x 5 x 2 4 3x _ 2 x 2 4 4. x 5 3x 2 _ x 2 4 4x 3 _ x 2 4 2. 7x _ 5 x 3 4x _ 1 x 3 7x 5 4x 1 x 3 3x 4 _ x 3 3. 2x _ 1 x 1 5x _ 4 x 1 2x 1 5x 4 x 1 3x 5 x 1 x 2, 2 x 3 x 1 4x 1 _ 3x 7 x _ 9 x 3x 7 3x 10 3x 7 5. 8 x x 3 5 x x 3 3 x 3 The denominators are the same. Subtract the numerators. Use the Distributive Property. Combine like terms. 6. 5x 2 _ x 2 1 3x _ 7 x 2 1 2x 9 _ x 2 1 7 3 x 3 x 1 22 Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions (continued) Use the least common denominator (LCD) to add rational expressions with different denominators. The process is the same as adding fractions with different denominators. Add: x 4 x 2 2x 3 2x x 1. Step 1 Factor denominators completely. x 4 x 2 2x 3 2x x 1 x 4 x 3 x 1 2x x 1 Step 2 Find the LCD. The LCD is the least common multiple of the denominators: x 3 x 1. Step 3 Write each term of the expression using the LCD. 2x x 1 2x x 1 x 3 x 3 2 x 2 6x x 1 x 3 So, x 4 x 3 x 1 2x x 1 x 4 Step 4 Add the numerators and simplify. x 4 2 x 2 6x 2 x 2 7x 4 x 3 x 1 x 3 x 1 x 3 x 1 Step 5 Identify x-values for which the expression is undefined. 2 x 2 6x x 1 x 3 x 3 or 1 because both values make the denominator equal 0. Add. 7. _ x 1 x 2 4 3x x 2 x 1 x 2 x 2 3x x 2 x 1 x 2 x 2 3x x 2 x 2 x 1 3 x 2 6x x 2 x 2 8. 4x 1 x 2 3x 2 3 x 1 4x 1 x 2 x 1 3 x 1 x 2 x 2 4x 1 3x 6 x 2 x 2 x 1 3 x 2 5x 1 7x 5 x 2 x 2 x 2 x 1 x 2, 2 x 2, 1 9. What is the LCD of 2x _ 1 x 2 9 and 7 x 2 x 6? x 3 x 3 x 2 23 Holt Algebra 2
8-4 Rational Functions A rational function can be written as a ratio of two polynomials. This is a rational function. f x a x h k The graph of this function is a hyperbola. There is a vertical asymptote at x h and the domain is { x x h }. There is a horizontal asymptote at y k and the range is { y y k }. Identify h and k to graph rational functions of the form f x a k. x h Graph g x h 2 3. x 2 Vertical asymptote at x 2. Horizontal asymptote at y 3. k 3 The graph of f x 1 x is translated 2 units right and 3 units down. Identify the asymptotes of each function. Describe the transformation of f x 1 x. Then graph each function. 1. g x x 1 2 2. g x x 1 3 Vertical asymptote: x 1 Vertical asymptote: x 1 Horizontal asymptote: y 2 Horizontal asymptote: y 3 translated 1 unit left and 2 units down translated 1 unit right and 3 units up 30 Holt Algebra 2
8-4 Rational Functions (continued) Use the zeros and the asymptotes of f x p x q x to graph f x. The zeros of f x occur where p x 0. The vertical asymptotes of f x occur where q x 0. The GCF of p x and q x must be 1. Graph f x x 2 2x 8. x 1 Step 1 Find the zeros. Factor the numerator: x 2 2x 8 x 2 x 4. The zeros occur at 2 and 4. Step 2 Find the vertical asymptotes. x 1 0 at x 1 Step 3 Graph. Plot the zeros at 2, 0 and 4, 0. Draw the vertical asymptote at x 1. Make a table of values and plot. x 8 0 2 10 y 8 8 8 8 x 2 0, so x 2 x 4 0, so x 4 Identify the zeros and the vertical asymptotes of each function. Then graph. 3. f x x 2 x 12 x 2 4. f x x 2 x 6 x 1 f x x 3 x 4 f x x 3 x 2 x 2 x 1 Zeros: 3, 4 Zeros: 3, 2 Vertical asymptote: x 2 Vertical asymptote: x 1 31 Holt Algebra 2
8-4 Practice C Rational Functions Identify the asymptotes, domain, and range of each function. 1. g x 2. g x x 5 7 4 x 9 1 4 3. g x x 2 12 3 Identify the zeros and asymptotes of the function. Then graph. 4. f x 2 x 2 18 x 2 25 a. Zeros: 3 and 3 b. Vertical asymptote: x 5 and x 5 c. Horizontal asymptote: d. Graph. y 2 Vertical asymptote: x 5; horizontal asymptote: y 7; domain: { x x 5 } ; range: { y y 7 } Vertical asymptote: x 9; horizontal asymptote: y 1 4 ; domain: { x x 9 } ; range: { y y 1 4 } Vertical asymptote: x 2 3 ; horizontal asymptote: y 12; domain: { x x 2 ; range: { y y 12 } 3} Identify holes in the graph of the function. Then graph. 5. f x x 2 2x 3 x 3 Solve. At x 3 6. The annual transportation costs, C, incurred by a company follow the formula C 2500 s s, where C is in thousands of dollars and s is the average speed the company s trucks are driven, in miles per hour. Use your graphing calculator to find the speed at which cost is at a minimum. 50 miles per hour 29 Holt Algebra 2