3.9 Multiplying & Dividing Rational Expressions p

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

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 x 4 = 12x 4 36x 4 = 2x22x2 3x 6x26x2 12x 3 = (2x 2 )(6x 2 ) (3x)(12x 3 )

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

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)

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).

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

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

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

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.

EXAMPLE 5 Divide a rational expression by a polynomial Find the quotient 2x x x23x2 (x +6). 2x x x23x2 (x +6) = 2x x x23x2 (x +6). = 1 Rewrite polynomial as fraction. 2x x x23x2 = 1 (x +6). Multiply by multiplicative inverse. 2x x x23x2 = (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.

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

EXAMPLE 6 Solve a multi-step problem A = 13, x 1 – 0.015x and T = x 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 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 ADVERTISING

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

EXAMPLE 6 Solve a multi-step problem x 1 – 0.016x13, x+ 1– 0.015x = ( x)(1 –0.015x) (1– 0.016x)(13, x) = Multiply by multiplicative inverse. Multiply numerators and denominators. Factor and divide out common factor. ( x)(1 – 0.015x) (1 – 0.016x)( x) = Simplify. 20 ( x) (1 – 0.015x) ( 1 – 0.016x)(20)( x) =

EXAMPLE 6 Solve a multi-step problem STEP 3 Approximate the percent spent on television advertising in 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 ( )(1 – ) (1 – )( ) P =P = 0.239

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 The answers are the same.