1. Multiply Polynomials
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
1. Simplify –2 (9a – b).
ANSWER
–18a + 2b
2. Simplify r2s rs3.
ANSWER
r3s4
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. Find the product 2x3(x3 + 3x2 – 2x + 5).
Example 1
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
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
STEP 2
Make a table of products.
3x2 x
– 4 3x 2
3x2 x
– 4 3x 2 2x
– 12x
– 8
ANSWER
The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8.

6. Guided Practice
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
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
STEP 3
Add products.
b2 + 6b – 7
3b – 4
– 4b2 – 24b + 28
3b3 + 18b2 – 21b
3b3 + 14b2 – 45b + 28

9. Find the product (2x2 + 5x – 1)(4x – 3).
Example 4
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. Example 4
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.
First
Outer
Inner
Last
(2x + 3)(4x + 1)
= 8x2
+ 2x
+ 12x
+ 3

11. Find the product (3a + 4)(a – 2).
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.
= 3a2 + (– 6a) + 4a + (– 8)
Multiply.
= 3a2 – 2a – 8
Combine like terms.

12. Guided Practice
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

13. Example 6
The dimensions of a rectangle are x + 3 and x 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.
= x2 + 5x + 6
Combine like terms.
ANSWER
The correct answer is B. A D C B

14. Example 6
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.

15. Guided Practice
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 C
ANSWER C

16. Example 7
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? •

17. Example 7
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.

18. Example 7
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.

19. Guided Practice
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.
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. Lesson Quiz
Find the product.
x(x3 – 3x2 + 2x – 4)
3x4 – 9x3 + 6x2 – 12x
ANSWER
2. (y – 4)(2y + 5)
ANSWER
2y2 – 3y – 20
3. (4x + 3)(3x – 2)
ANSWER
12x2 + x – 6
4. (b2 – 2b – 1)(3b – 5)
ANSWER
3b3 – 11b2 + 7b + 5

21. Lesson 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