1. 1. Simplify –2 (9a – b).
ANSWER
–18a + 2b
2. Simplify r2s rs3.
ANSWER
r3s4

2. 3. The number of hardback h and paperback p books (in
3. The number of hardback h and paperback p books (in hundreds) sold from 1999–2005 can be modeled by
h = 0.2t2 – 1.7t + 14 and p = 0.17t3 – 2.7t t + 27 where t is the number of years since About how many books sold in 2003?
ANSWER
5200

3. Multiply a monomial and a polynomial
EXAMPLE 1
Multiply a monomial and a polynomial
Find the product 2x3(x3 + 3x2 – 2x + 5).
2x3(x3 + 3x2 – 2x + 5)
Write product.
= 2x3(x3) + 2x3(3x2) – 2x3(2x) + 2x3(5)
Distributive property
= 2x6 + 6x5 – 4x4 + 10x3
Product of powers property

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

5. EXAMPLE 2
Multiply polynomials using a table
STEP 2
Make a table of products.
3x2 x
– 4 3x 2
– 8
– 12x 2x
3x2 x
– 4 3x 2
ANSWER
The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8.

6. GUIDED PRACTICE
for Examples 1 and 2
Find the product.
1. x(7x2 +4)
ANSWER
7x3 + 4x
2. (a +3)(2a +1)
ANSWER
2a2 + 7a + 3
ANSWER
4n2 + 19n – 5
3. (4n – 1)(n + 5)

7. EXAMPLE 3
Multiply polynomials vertically
Find the product (b2 + 6b – 7)(3b – 4).
SOLUTION
STEP 1
Multiply by – 4.
STEP 2
Multiply by 3b.
b2 + 6b – 7
b2 + 6b – 7
3b – 4
3b – 4
– 4b2 – 24b + 28
– 4b2 – 24b + 28
3b3 + 18b2 – 21b

8. EXAMPLE 3
Multiply polynomials vertically
STEP 3
Add products.
b2 + 6b – 7
3b – 4
– 4b2 – 24b + 28
3b3 + 18b2 – 21b
3b3 + 14b2 – 45b + 28

9. Multiply polynomials horizontally
EXAMPLE 4
Multiply polynomials horizontally
Find the product (2x2 + 5x – 1)(4x – 3).
(2x2 + 5x – 1)(4x – 3)
Write product.
= 2x2(4x – 3) + 5x(4x – 3) – 1(4x – 3)
Distributive property
= 8x3 – 6x2 + 20x2 – 15x – 4x + 3
Distributive property
= 8x3 + 14x2 – 19x + 3
Combine like terms.

10. Multiply binomials using the FOIL pattern
EXAMPLE 5
Multiply binomials using the FOIL pattern
Find the product (3a + 4)(a – 2).
(3a + 4)(a – 2)
= (3a)(a) + (3a)(– 2) + (4)(a) + (4)(– 2)
Write products of terms.
= 3a2 + (– 6a) + 4a + (– 8)
Multiply.
= 3a2 – 2a – 8
Combine like terms.

11. GUIDED PRACTICE
for Examples 3, 4, and 5
Find the product.
(x2 + 2x +1)(x + 2) 4.
x3 + 4x2 + 5x + 2
ANSWER
(3y2 –y + 5)(2y – 3)
ANSWER
6y3 – 11y2 + 13y – 15
(4b –5)(b – 2)
ANSWER
4b2 – 13b + 10

12. Standardized Test Practice
EXAMPLE 6
Standardized Test Practice
The dimensions of a rectangle are x + 3 and x + 2. Which expression represents the area of the rectangle?
x2 + 6 A
x2 + 5x + 6 B
x2 + 6x + 6 C
x2 + 6x D
SOLUTION
Area = length width
Formula for area of a rectangle
= (x + 3)(x + 2)
Substitute for length and width.
= x2 + 2x + 3x + 6
Multiply binomials.

13. Standardized Test Practice
EXAMPLE 6
Standardized Test Practice
= x2 + 5x + 6
Combine like terms.
ANSWER
The correct answer is B. A D C B
CHECK
You can use a graph to check your answer. Use a graphing calculator to display the graphs of y1 = (x + 3) (x + 2) and y2 = x2 + 5x + 6 in the same viewing window. Because the graphs coincide, you know that the product of x + 3 and x + 2 is x2 + 5x + 6.

14. EXAMPLE 7
Solve a multi-step problem
SKATEBOARDING
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? •

15. Solve a multi-step problem
EXAMPLE 7
Solve a multi-step problem
SOLUTION
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.
= – 45x – 33x + x2
Multiply binomials.
= 1485 – 78x + x2
Combine like terms.

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

17. GUIDED PRACTICE
for Examples 6 and 7
The dimensions of a rectangle are x + 5 and x + 9. Which expression represents the area of the rectangle? 7.
x2 + 45x A
x2 + 45 B
x2 + 14x + 45 C
x2 + 45x + 45 D
ANSWER C

18. GUIDED PRACTICE
for Examples 6 and 7 8.
GARDEN DESIGN
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.

19. GUIDED PRACTICE
for Examples 6 and 7
a. Write a polynomial that represents the combined area of the garden and the walkway.
4x2 + 38x + 90
ANSWER
b. Find the combined area when the width of the walkway is 4 feet.
306 ft2
ANSWER

20. Warm-up: Read and take notes starting on page 503
Warm-up: Read and take notes starting on page 503. Try and complete #1-6 Homework: Worksheet given in class.

21. Warm-up: Homework: Page 508 #17-42 all

22. Daily Homework Quiz
Find the product.
x(x3 – 3x2 + 2x – 4)
3x4 – 9x3 + 6x2 – 12x
ANSWER
2. (y – 4)(2y + 5)
ANSWER
2y2 – 3y – 20

23. Daily Homework Quiz
3. (4x + 3)(3x – 2)
ANSWER
12x2 + x – 6
4. (b2 – 2b – 1)(3b – 5)
ANSWER
3b3 – 11b2 + 7b + 5

24. 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
3x2 + 11x – 4