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# 11.3

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### 11.3

1. 1. Multiplying and Dividing RationalExpressionsLet a, b, c, and d be polynomials. a c ac whereAlgebra: × = b ¹ 0 and d ¹ 0 b d bd a c a d ad where b ¹ 0, c ¹ 0, and d ¹ 0 ¸ = × = b d b c bcExamples: x + 2 × 32 = 3(x + 2) x x x3 x 4 x x x2 ¸ = × = x -1 x x -1 4 4(x -1)
2. 2. Example 1 Multiply rational expressions involving monomials 2x 2 6x 2Find the product • 4 . 3x 12x2x 2 6x 2 (2x 2) (6x 2) Multiply numerators and • 4 = denominators.3x 12x (3x ) (12x 4 ) 12x 4 = Product of powers property 36x 5 12 • x 4 Factor and divide out common = factors. 3 • 12 • x 4 • x 1 = Simplify. 3x
3. 3. Example 1 Multiply rational expressions involving monomialsCHECK Check your simplification using a graphing calculator. 2x 2 6x 2 Graph y1 = • 4 and 3x 12x 1 y2 = . 3x The graphs coincide. So, the expressions are equivalent for all values of x other than 0.
4. 4. Example 2 Multiple Choice Practice x 2 – 4x + 3 3x 2 + 3xWhat is the product of and ? 2 – x 4x 2 – 24x + 36 x 3x 3 ( x – 1) 3 ( x + 1) x+1 4 ( x – 3) – x 3 4 ( x + 3) x +3SOLUTION 3x 2 + 3x x 2 – 4x + 3 • 4x 2 – 24x + 36 x2 – x ( 3x 2 + 3x ) ( x 2 – 4x + 3) Multiply numerators and = denominators. (4x 2 – 24x + 36)( x 2 – x)
5. 5. Example 2 Multiple Choice Practice 3x ( x + 1) ( x – 3) ( x – 1) Factor and divide out = 4x ( x – 3) ( x – 3) ( x – 1) common factors. 3 ( x + 1) = Simplify. 4 ( x – 3)ANSWER The correct answer is C.
6. 6. Example 3 Multiply a rational expression by a polynomial 5xFind the product 2 • ( x + 3). x + 5x + 6 5x 2 + 5x • ( x + 3)x + 6 5x x+3 Rewrite polynomial as a = 2 • x + 5x + 6 1 fraction. 5x ( x + 3) Multiply numerators and = 2 denominators. x + 5x + 6 5x ( x + 3) Factor and divide out common = factor. ( x + 2) ( x + 3)
7. 7. Example 3 Multiply a rational expression by a polynomial 5x = Simplify. x+2
8. 8. Example 4 Divide rational expressions involving polynomials 7x 2 – 7x x +1Find the quotient 2 ÷ 2 . x + 2x – 3 x – 7x – 8 7x 2 – 7x x +1 2 + 2x – 3 ÷ 2x x – 7x – 8 7x 2 – 7x x 2 – 7x – 8 Multiply by multiplicative = 2 • x + 2x – 3 x +1 inverse. (7x 2 – 7x) ( x 2 – 7x – 8 ) Multiply numerators and = ( x 2 + 2x – 3) ( x + 1) denominators. 7x ( x – 1) ( x – 8) ( x + 1) Factor and divide out = ( x + 3) ( x – 1) ( x + 1) common factors.
9. 9. Example 4 Divide rational expressions involving polynomials 7x ( x – 8) Simplify. = x +3
10. 10. Example 5 Divide a rational expression by a polynomial 2x 2 + 16x + 24Find the quotient 2 ÷ ( x + 6 ). 3x2x 2 + 16x + 24 2 ÷ ( x + 6) 3x 2x 2 + 16x + 24 x +6 Rewrite polynomial as = ÷ 3x 2 1 fraction. 2x 2 + 16x + 24 1 Multiply by multiplicative = • inverse. 3x 2 x +6 2x 2 + 16x + 24 Multiply numerators and = denominators. 3x 2 ( x + 6 )
11. 11. Example 5 Divide a rational expression by a polynomial 2 (x + 2) (x + 6) Factor and divide out = common factor. 3x 2 ( x + 6 ) 2 (x + 2) = Simplify. 3x 2
12. 12. Example 6 Solve a multi-step problemBASEBALLHank Aaron’s career number B of times at bat andcareer number H of hits through each year of theperiod 1954–1976 can be modeled by 300 + 700x 62 + 240x B = and H = 1 + 0.01x 1 + 0.017xwhere x is the number of years since 1954. A baseballplayer’s batting average is the number of hits dividedby the number of times at bat.• Write a model that gives Hank Aaron’s career batting average A as a function of x.
13. 13. Example 6 Solve a multi-step problem• Approximate his career batting average in 1959.SOLUTION STEP 1 Write a verbal model. Then write an equation. = ÷ A = H ÷ B STEP 2 Find the quotient. A =H ÷B Write equation.
14. 14. Example 6 Solve a multi-step problem 62 + 240x 300 + 700x = ÷ Substitute for H and B. 1 + 0.017x 1 + 0.01x 62 + 240x 1 + 0.01x = • Multiply by multiplicative 1 + 0.017x 300 + 700x inverse. ( 62 + 240x )( 1 + 0.01x ) Multiply numerators and = ( 1 + 0.017x )( 300 + 700x ) denominators. (2)( 31 + 120x )( 1 + 0.01x ) = Factor and divide out ( 1 + 0.017x ) (2)( 150 + 350x ) common factor.
15. 15. Example 6 Solve a multi-step problem ( 31 + 120x )( 1 + 0.01x ) Simplify. = (1 + 0.017x ) (150 + 350x )STEP 3 Approximate Aaron’s career batting average in 1959. Substitute 5 for x in the model and use a calculator to evaluate. ( 31 + 120 • 5 )( 1 + 0.01 • 5 ) A = ≈ .321 ( 1 + 0.017 • 5)( 150 + 350 • 5 )ANSWERAccording to the model, Hank Aaron’s career battingaverage in 1959 was approximately .321.
16. 16. 11.3 Warm-Up