1. 9.2 Multiply Polynomials I can…multiply polynomials
Students will do assigned homework.
Students will study vocabulary words.

2. Daily Homework Quiz
For use after Lesson 9.1
If the expression is a polynomial, find its degree and classify it by the number of terms. Otherwise, tell why it is not a polynomial.
1. m3 + n4m2 + m–2
No; one exponent is not a whole number.
ANSWER
2. – 3b3c4 – 4b2c + c8
ANSWER
8th degree trinomial

3. Daily Homework Quiz
For use after Lesson 9.1
Find the sum or difference.
3. (3m2 – 2m + 9) + (m2 + 2m – 4)
4m2 + 5
ANSWER
4. (– 4a2 + 3a – 1) – (a2 + 2a – 6)
ANSWER
–5a2 + a + 5

4. 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

5. GUIDED PRACTICE
for Examples 1 and 2
Find the product.
1. x(7x2 +4)
ANSWER
7x3 + 4x
2. 3a(2a +1)
ANSWER
3a2 + 3a
ANSWER
4n2 + 20n
3. 4n (n + 5)

6. 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)

7. 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.

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

9. 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

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

11. 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.
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.

12. 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
(4b2 –5b + 6)(b – 2)
ANSWER
4b2 – 13b2 + 16b – 12

13. Multiply polynomials horizontally
EXAMPLE 4
Multiply polynomials horizontally
First
Outer
Inner
Last
(2x + 3)(4x + 1)
= 8x2 + 2x + 12x + 3
= (2x)(4x) + (2x)(1) + (3)(4x) + (3)(1)
Write products of terms.
= 8x2 + 2x + 12x + 3
Multiply.
= 8x2 + 14x + 3
Combine like terms.

14. Multiply binomials using the FOIL pattern
EXAMPLE 5
Multiply binomials using the FOIL pattern
Find the product (3a + 4)(a – 2).
First
Outer
Inner
Last
(3a + 4)( a - 2)
= 3a2 – 6a + 4a – 8
(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.

15. GUIDED PRACTICE
for Examples 3, 4, and 5
Find the product.
(x + 3)(x + 2) 4.
x2 + 5x + 6
ANSWER
(y + 5)(2y – 3)
ANSWER
2y2 + 7y - 15
(4b –5)(b – 2)
ANSWER
4b2 – 13b + 10

16. 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.

17. 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.

18. 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? •

19. 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.

20. 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.

21. 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 C
ANSWER C

22. 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.

23. 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