9.2 Multiply Polynomials

•I can…multiply polynomials

•Students will do assigned homework.

•Students will study vocabulary words.

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

Daily Homework Quiz

For use after Lesson 9.1

Find the sum or difference.

3. (3m2 – 2m + 9) + (m2 + 2m – 4)

4m2 + 5ANSWER

4. (– 4a2 + 3a – 1) – (a2 + 2a – 6)

ANSWER –5a2 + a + 5

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

GUIDED PRACTICE for Examples 1 and 2

Find the product.

1. x(7x2 +4) ANSWER 7x3 + 4x

2. 3a(2a +1) ANSWER 3a2 + 3a

3. 4n (n + 5) ANSWER 4n2 + 20n

Multiply polynomials using a tableEXAMPLE 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

Multiply polynomials using a tableEXAMPLE 2

ANSWER

The product is 3x2 + 2x – 12x – 8, or 3x2 – 10x – 8.

3x2x

– 4

3x 2

– 8– 12x

2x3x2x

– 4

3x 2

STEP 2

Make a table of products.

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

3. (4n – 1)(n + 5) ANSWER 4n2 + 19n – 5

EXAMPLE 3 Multiply polynomials vertically

Find the product (b2 + 6b – 7)(3b – 4).

SOLUTION

STEP 1 Multiply by – 4.

b2 + 6b – 7

– 4b2 – 24b + 28

3b – 4

STEP 2Multiply by 3b.

b2 + 6b – 7

3b – 4

– 4b2 – 24b + 28 3b3 + 18b2 – 21b

Multiply polynomials vertically

EXAMPLE 3

STEP 3

Add products.

b2 + 6b – 7

3b – 4

– 4b2 – 24b + 28

3b3 + 18b2 – 21b

3b3 + 14b2 – 45b + 28

Multiply polynomials horizontallyEXAMPLE 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)

= 8x3 – 6x2 + 20x2 – 15x – 4x + 3

Distributive property

Distributive property

= 8x3 + 14x2 – 19x + 3 Combine like terms.

FOIL PATTERN The letters of the word FOIL can helpyou to remember how to use the distributive property tomultiply binomials. The letters should remind you of thewords First, Outer, Inner, and Last.

GUIDED PRACTICE for Examples 3, 4, and 5

Find the product.

(x2 + 2x +1)(x + 2)4. x3 + 4x2 + 5x + 2ANSWER

5. (3y2 –y + 5)(2y – 3) ANSWER 6y3 – 11y2 + 13y – 15

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

Multiply polynomials horizontally

EXAMPLE 4

(2x + 3)(4x + 1)

First Outer Inner Last

= 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.

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.

= 3a2 + (– 6a) + 4a + (– 8) Multiply.

= 3a2 – 2a – 8 Combine like terms.

(3a + 4)( a - 2)

First Outer Inner Last

= 3a2 – 6a + 4a – 8

GUIDED PRACTICE for Examples 3, 4, and 5

Find the product.

(x + 3)(x + 2)4. x2 + 5x + 6ANSWER

5. (y + 5)(2y – 3) ANSWER 2y2 + 7y - 15

6. (4b –5)(b – 2) ANSWER 4b2 – 13b + 10

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.

= x2 + 2x + 3x + 6 Multiply binomials.

x2 + 6 A x2 + 6xDx2 + 5x + 6B x2 + 6x + 6C

EXAMPLE 6

= x2 + 5x + 6 Combine like terms.

ANSWER

The correct answer is B. A DCB

Standardized Test Practice

CHECKYou 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.

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?

•

SOLUTION

EXAMPLE 7 Solve a multi-step problem

STEP 1Write 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 + x2 Multiply binomials.

= 1485 – 78x + x2 Combine like terms.

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.

GUIDED PRACTICE for Examples 6 and 7

x2 + 45x A

x2 + 45x + 45D

x2 + 45B

x2 + 14x + 45C

The dimensions of a rectangle are x + 5 and x + 9. Which expression represents the area of the rectangle?

7

ANSWER C

C

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

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.

4x2 + 38x + 90ANSWER

306 ft2ANSWER