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3.9 Multiplying & Dividing Rational Expressions p. 165-168

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Multiplying Factor numerator and denominator of each expression Combine both numerators on top Combine both denominators on bottom Look to cancel any common factors Simplify

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Multiply rational expressions involving monomials EXAMPLE 1 Find the product 2x22x2 3x 6x 2 12x 3 Multiply numerators and denominators. Product of powers property Factor and divide out common factors. Simplify. 1 3 = 12 x 4 3 12 x 4 = 12x 4 36x 4 = 2x22x2 3x 6x26x2 12x 3 = (2x 2 )(6x 2 ) (3x)(12x 3 )

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GUIDED PRACTICE for Example 1 Find the product 2y32y3 5y 15y 3 8y 5 1. 3 4 ANSWER 2. 7z27z2 4z 3 z 3 14z ANSWER z 8

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EXAMPLE 2 Multiply rational expressions involving polynomials Find the product 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x Multiply numerators and denominators. 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x = (3x 2 + 3x) (x 2 – 4x + 3) (4x 2 – 24x + 36)(x 2 – x) Factor and divide out common factors. = 3(x + 1) 4(x – 3) Simplify. = 3x(x + 1)(x – 3)(x – 1) 4x(x – 3)(x – 3)(x – 1)

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EXAMPLE 2 Multiply rational expressions involving polynomials CHECK Check your simplification using a graphing calculator. The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3). Graph y 1 = 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x and y 2 = 3(x + 1) 4(x – 3).

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EXAMPLE 3 Multiply a rational expression by a polynomial Find the product 5x5x x 2 + 5x + 6 (x + 3). 5x5x x 2 + 5x + 6 (x + 3). 5x5x x 2 + 5x + 6 (x + 3) = 1 Rewrite polynomial as a fraction. = 5x(x + 3) x 2 + 5x + 6 Multiply numerators and denominators. = 5x(x + 3) (x + 3)(x + 2) Factor and divide out common factor. Simplify. = 5x5x x + 2

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GUIDED PRACTICE for Examples 2 and 3 3. Find the product x 2 + x – 2 x 2 + x 2x 2 + 2x 5x 2 –15x +10 2(x + 2) 5(x – 2) ANSWER Find the product 2w22w2 w 2 – 7w + 12 (w – 4). 4. 2w22w2 w – 3 ANSWER

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Dividing Multiply by the multiplicative inverse (reciprocal) of the second expression Factor & write as one rational expression Cancel/divide out common factors Simplify

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EXAMPLE 4 Divide rational expressions involving polynomials Find the quotient. 7x 2 – 7x x 2 + 2x – 3 x + 1 x 2 – 7x – 8 7x 2 – 7x x 2 + 2x – 3 x + 1 x 2 – 7x – 8 7x 2 – 7x x 2 + 2x – 3x + 1 = x 2 – 7x – 8 Multiply by multiplicative inverse. = (7x 2 – 7x) (x 2 – 7x – 8) (x + 1)(x 2 + 2x – 3) Multiply numerators and denominators. = 7x(x – 1)(x – 8)(x + 1) (x + 3)(x – 1)(x + 1) Factor and divide out common factors. = 7x(x – 8) x + 3 Simplify.

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EXAMPLE 5 Divide a rational expression by a polynomial Find the quotient 2x 2 + 16x + 24 3x23x2 (x +6). 2x 2 + 16x + 24 3x23x2 (x +6) = 2x 2 + 16x + 24 3x23x2 (x +6). = 1 Rewrite polynomial as fraction. 2x 2 + 16x + 24 3x23x2 = 1 (x +6). Multiply by multiplicative inverse. 2x 2 + 16x + 24 3x23x2 = (x +6). Multiply numerators and denominators. = 2(x + 2)(x + 6) 3x 2 (x + 6) Factor and divide out common factor. = 2(x + 2) 3x23x2 Simplify.

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GUIDED PRACTICE for Examples 4 and 5 5. m 2 – 4 2m 2 + 4m 6m – 3m 2 4m + 44 2 (m +11) 3m23m2 – ANSWER 6. n 2 – 6n + 9 12n n – 3 ANSWER 12n n – 3 Find the quotient.

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EXAMPLE 6 Solve a multi-step problem A = 13,000 + 3700x 1 – 0.015x and T = 1800 + 860x 1 – 0.016x The amount A (in millions of dollars) spent on all advertising and the amount T (in millions of dollars) spent on television advertising in the United States during the period 1970–2003 can be modeled by where x is the number of years since 1970. Write a model that gives the percent p (in decimal form) of the amount spent on all advertising that was spent on television advertising. Then approximate the percent spent on television advertising in 2003. ADVERTISING

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EXAMPLE 6 Solve a multi-step problem STEP 1 SOLUTION Write a verbal model. Then write an equation. P= TA p =TA Write equation. 1800 + 860x 1 – 0.016 13,000 + 3700x 1 – 0.015x x = STEP 2 Find the quotient. Substitute for T and for A.

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EXAMPLE 6 Solve a multi-step problem 1800 + 860x 1 – 0.016x13,0003700x+ 1– 0.015x = (1800 + 860x)(1 –0.015x) (1– 0.016x)(13,000 + 3700x) = Multiply by multiplicative inverse. Multiply numerators and denominators. Factor and divide out common factor. (90 + 43x)(1 – 0.015x) (1 – 0.016x)(650 + 185x) = Simplify. 20 (90 + 43x) (1 – 0.015x) ( 1 – 0.016x)(20)(650 + 185x) =

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EXAMPLE 6 Solve a multi-step problem STEP 3 Approximate the percent spent on television advertising in 2003. Because 2003 – 1970 = 33, x = 33. Substitute 33 for x in the model and use a calculator to evaluate. ANSWER About 24% of the amount spent on all advertising was spent on television advertising in 2003. (90 + 43 33)(1 – 0.015 33) (1 – 0.016 33)(650 + 185 33) P =P = 0.239

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GUIDED PRACTICE for Example 6 7. In Example 6, find the values of T and of A separately when x = 33. Then divide the value of T by the value of A. Compare your answer with the answer in Step 3 above. ANSWER About $63,941 million About $267,525 million About 0.239 The answers are the same.

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