Warm Up Lesson Presentation Lesson Quiz Multiply and Divide Rational Expressions Warm Up Lesson Presentation Lesson Quiz

Warm-Up Factor the polynomial. 1. x2 + 3x – 18 ANSWER (x – 3)(x + 6) 2. 2x2 – 4x – 16 ANSWER 2(x – 4)(x + 2) 3. 2x2 + 8x – 10 ANSWER 2(x + 5)(x – 1)

Warm-Up 4. A golf clinic charges a $30 enrollment fee and $5 per session. What is the average cost per session if you attend 8 sessions? ANSWER $8.75

Example 1 6x2 2x2 Find the product . 12x3 3x (2x2)(6x2) 6x2 2x2 = 12x3 Multiply numerators and denominators. 12x4 36x4 = Product of powers property 12 x4 3 12 x4 = Factor and divide out common factors. 1 3 = Simplify.

Guided Practice Find the product 2y3 5y 15y3 8y5 1. 3 4 ANSWER 2. 7z2 4z3 z3 14z ANSWER z 8

Example 2 3x2 + 3x 4x2 – 24x + 36 x2 – 4x + 3 x2 – x Find the product = (3x2 + 3x) (x2 – 4x + 3) (4x2 – 24x + 36)(x2 – x) Multiply numerators and denominators. = 3x(x + 1)(x – 3)(x – 1) 4x(x – 3)(x – 3)(x – 1) Factor and divide out common factors. = 3(x + 1) 4(x – 3) Simplify.

Example 2 CHECK Check your simplification using a graphing calculator. Graph y1 = 3x2 + 3x 4x2 – 24x + 36 x2 – 4x + 3 x2 – x and y2 = 3(x + 1) 4(x – 3) . The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3).

Guided Practice 3. Find the product x2 + x – 2 x2 + x 2x2 + 2x 5x2 –15x +10 2(x + 1) 5(x – 2) ANSWER

Example 3 Find the product 5x x2 + 5x + 6 (x + 3). 5x x2 + 5x + 6 = 1 Rewrite polynomial as a fraction. = 5x(x + 3) x2 + 5x + 6 Multiply numerators and denominators. = 5x(x + 3) (x + 3) (x + 2) Factor and divide out common factor. = 5x x + 2 Simplify.

Guided Practice Find the product 2w2 w2 – 7w + 12 (w – 4). 4. 2w2 w – 3 ANSWER

Example 4 Find the quotient . 7x2 – 7x x2 + 2x – 3 x + 1 x2 – 7x – 8 = x2 – 7x – 8 Multiply by multiplicative inverse. = (7x2 – 7x) (x2 – 7x – 8) (x + 1) (x2 + 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.

Guided Practice Find the quotient. 5. m2 – 4 2m2 + 4m 6m – 3m2 4m + 44 2(m + 11) 3m2 – ANSWER

Example 5 Find the quotient 2x2 + 16x + 24 3x2 (x + 6). 2x2 + 16x + 24 = 2x2 + 16x + 24 3x2 (x + 6). = 1 Rewrite polynomial as fraction. 2x2 + 16x + 24 3x2 = 1 (x + 6). Multiply by multiplicative inverse. 2x2 + 16x + 24 3x2 = (x + 6). Multiply numerators and denominators. = 2(x + 2)(x + 6) 3x2(x + 6) Factor and divide out common factor. = 2(x + 2) 3x2 Simplify.

Guided Practice Find the quotient. 6. n2 – 6n + 9 12n n – 3 ANSWER 12n n – 3

Example 6 ADVERTISING 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 A = 13,000 + 3700x 1 – 0.015x and T = 1800 + 860x 1 – 0.016x 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.

Write a verbal model. Then write an equation. Example 6 SOLUTION STEP 1 Write a verbal model. Then write an equation. p = T A STEP 2 Find the quotient. p = T A Write equation. 1800 + 860x 1 – 0.016 13,000 3700x 0.015x x = Substitute for T and for A.

Example 6 1800 + 860x 1 – 0.016 x 13,000 3700x 1– 0.015x = Multiply by multiplicative inverse. (1800 + 860x)(1 – 0.015x) (1– 0.016x)(13,000 + 3700x) = Multiply numerators and denominators. 20 (90 + 43x) (1 – 0.015x) (1 – 0.016x)(20)(650 + 185x) = Factor and divide out common factor. (90 + 43x)(1 – 0.015x) (1 – 0.016x)(650 + 185x) = Simplify.

Example 6 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. (90 + 43 33)(1 – 0.015 33) (1 – 0.016 33)(650 + 185 33) P = 0.239 ANSWER About 24% of the amount spent on all advertising was spent on television advertising in 2003.

Guided Practice 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.

Lesson Quiz 1. Find the product 3x x2 – 3x – 10 (x + 2) ANSWER 3x x – 5 2. Find the quotient ( 2x3 + 11x2 + 11x – 4 ) ( x + 4) ANSWER 2x2 + 3x – 1

Lesson Quiz 3. The average total revenue (in thousands of dollars) of books B sold at a used bookstore and the average revenue of paper back books P sold at the store during 1990 – 2004 can be represented by and where x is the number of years since 1990. Approximate the percent of the total revenue of the books from paperback sales in 2004. 824 + 12x 1 – 0.02x B = 720 – 36x 1 – 0.017x P = ANSWER About 21%