1. Warm-Up Exercises 1. Simplify –2 (9a – b). ANSWER –18a + 2b ANSWER r3s4r3s4 2. Simplify r 2 s rs 3.

2. Warm-Up Exercises 3. The number of hardback h and paperback p books (inhundreds) sold from 1999–2005 can be modeled by h =0.2t 2 – 1.7t + 14 and p = 0.17t 3 – 2.7t 2 + 11.7t + 27 where t is the number of years since 1999. About how many books sold in 2003 ? ANSWER 5200

3. Warm-Up Exercises EXAMPLE 1 Multiply a monomial and a polynomial Find the product 2x 3 (x 3 + 3x 2 – 2x + 5). 2x 3 (x 3 + 3x 2 – 2x + 5) Write product. = 2x 3 (x 3 ) + 2x 3 (3x 2 ) – 2x 3 (2x) + 2x 3 (5) Distributive property = 2x 6 + 6x 5 – 4x 4 + 10x 3 Product of powers property

4. Warm-Up Exercises Multiply polynomials using a table EXAMPLE 2 Find the product (x – 4)(3x + 2). STEP 1 Write subtraction as addition in each polynomial. (x – 4)(3x + 2) = [x + (– 4)](3x + 2) SOLUTION

5. Warm-Up Exercises Multiply polynomials using a table EXAMPLE 2 ANSWER The product is 3x 2 + 2x – 12x – 8, or 3x 2 – 10x – 8. 3x23x2 x – 4 3x3x 2 – 8– 12x 2x2x3x23x2 x – 4 3x3x 2 STEP 2 Make a table of products.

6. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Find the product. 1. x(7x 2 +4) ANSWER 7x 3 + 4x 2. (a +3)(2a +1) ANSWER 2a 2 + 7a + 3 3. (4n – 1)(n + 5) ANSWER 4n 2 + 19n – 5

7. Warm-Up Exercises EXAMPLE 3 Multiply polynomials vertically Find the product (b 2 + 6b – 7)(3b – 4). SOLUTION STEP 1 Multiply by – 4. b 2 + 6b – 7 – 4b 2 – 24b + 28 3b – 4 STEP 2 Multiply by 3b. b 2 + 6b – 7 3b – 4 – 4b 2 – 24b + 28 3b 3 + 18b 2 – 21b

8. Warm-Up Exercises Multiply polynomials vertically EXAMPLE 3 STEP 3 Add products. b 2 + 6b – 7 3b – 4 – 4b 2 – 24b + 28 3b 3 + 18b 2 – 21b 3b 3 + 14b 2 – 45b + 28

9. Warm-Up Exercises Multiply polynomials horizontally EXAMPLE 4 Find the product (2x 2 + 5x – 1)(4x – 3). (2x 2 + 5x – 1)(4x – 3) Write product. = 2x 2 (4x – 3) + 5x(4x – 3) – 1(4x – 3) = 8x 3 – 6x 2 + 20x 2 – 15x – 4x + 3 Distributive property = 8x 3 + 14x 2 – 19x + 3 Combine like terms. FOIL PATTERN The letters of the word FOIL can help you to remember how to use the distributive property to multiply binomials. The letters should remind you of the words First, Outer, Inner, and Last.

10. Warm-Up Exercises Multiply polynomials horizontally EXAMPLE 4 (2x + 3)(4x + 1) First OuterInner Last = 8x 2 + 2x + 12x + 3

11. Warm-Up Exercises Multiply binomials using the FOIL pattern EXAMPLE 5 Find the product (3a + 4)(a – 2). (3a + 4)(a – 2) = (3a)(a) + (3a)(– 2) + (4)(a) + (4)(– 2) Write products of terms. = 3a 2 + (– 6a) + 4a + (– 8) Multiply. = 3a 2 – 2a – 8 Combine like terms.

12. Warm-Up Exercises GUIDED PRACTICE for Examples 3, 4, and 5 Find the product. (x 2 + 2x +1)(x + 2)4. x 3 + 4x 2 + 5x + 2 ANSWER 5. (3y 2 –y + 5)(2y – 3) ANSWER 6y 3 – 11y 2 + 13y – 15 6. (4b –5)(b – 2) ANSWER 4b 2 – 13b + 10

13. Warm-Up Exercises SOLUTION EXAMPLE 6 Standardized Test Practice The dimensions of a rectangle are x + 3 and x + 2. Which expression represents the area of the rectangle ? Area = length width Formula for area of a rectangle = (x + 3)(x + 2) Substitute for length and width. = x 2 + 2x + 3x + 6 Multiply binomials. x 2 + 6 A x 2 + 6x D x 2 + 5x + 6 B x 2 + 6x + 6 C

14. Warm-Up Exercises EXAMPLE 6 = x 2 + 5x + 6 Combine like terms. ANSWER The correct answer is B. A D C B Standardized Test Practice CHECK You can use a graph to check your answer. Use a graphing calculator to display the graphs of y 1 = (x + 3)(x + 2) and y 2 = x 2 + 5x + 6 in the same viewing window. Because the graphs coincide, you know that the product of x + 3 and x + 2 is x 2 + 5x + 6.

15. Warm-Up Exercises SKATEBOARDING EXAMPLE 7 Solve a multi-step problem You are designing a rectangular skateboard park on a lot that is on the corner of a city block. The park will have a walkway along two sides. The dimensions of the lot and the walkway are shown in the diagram. Write a polynomial that represents the area of the skateboard park. What is the area of the park if the walkway is 3 feet wide ?

16. Warm-Up Exercises SOLUTION EXAMPLE 7 Solve a multi-step problem STEP 1 Write a polynomial using the formula for the area of a rectangle. The length is 45 – x. The width is 33 – x. Area = length width Formula for area of a rectangle = (45 – x)(33 – x) Substitute for length and width. = 1485 – 45x – 33x + x 2 Multiply binomials. = 1485 – 78x + x 2 Combine like terms.

17. Warm-Up Exercises EXAMPLE 7 Solve a multi-step problem Substitute 3 for x and evaluate. STEP 2 Area = 1485 – 78(3) + (3) 2 = 1260 ANSWER The area of the park is 1260 square feet.

18. Warm-Up Exercises GUIDED PRACTICE for Examples 6 and 7 x 2 + 45x A x 2 + 45x + 45 D x 2 + 45 B x 2 + 14x + 45 C The dimensions of a rectangle are x + 5 and x + 9. Which expression represents the area of the rectangle ? 7 ANSWERC C

19. Warm-Up Exercises GUIDED PRACTICE for Examples 6 and 7 You are planning to build a walkway that surrounds a rectangular garden, as shown. The width of the walkway around the garden is the same on every side. 8. GARDEN DESIGN

20. Warm-Up Exercises GUIDED PRACTICE for Examples 6 and 7 a. Write a polynomial that represents the combined area of the garden and the walkway. b. Find the combined area when the width of the walkway is 4 feet. 4x 2 + 38x + 90 ANSWER 306 ft 2 ANSWER

21. Warm-Up Exercises Daily Homework Quiz Find the product. 1. 3x(x 3 – 3x 2 + 2x – 4) 3x 4 – 9x 3 + 6x 2 – 12x ANSWER 2. (y – 4)(2y + 5) ANSWER 2y 2 – 3y – 20

22. Warm-Up Exercises Daily Homework Quiz 3. (4x + 3)(3x – 2) ANSWER 12x 2 + x – 6 4. (b 2 – 2b – 1)(3b – 5) ANSWER 3b 3 – 11b 2 + 7b + 5

23. Warm-Up Exercises Daily Homework Quiz 5. The dimensions of a rectangle are x + 4 and 3x – 1. Write an expression to represent the area of the rectangle. ANSWER 3x 2 + 11x – 4