Algebra unit 8.2 8.3

UNIT 8.2-8.3 MULTIPLYING POLYNOMIALSUNIT 8.2-8.3 MULTIPLYING POLYNOMIALS

Warm UpEvaluate.

1. 32

3. 102

Simplify.

4. 23 • 24

6. (53)2

9 16

100

27

2. 24

5. y5 • y4

56 7. (x2)4

8. –4(x – 7) –4x + 28

y9

x8

Multiply polynomials.

Objective

To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

Multiply.

Example 1: Multiplying Monomials

A. (6y3)(3y5) (6y3)(3y5)

18y8

Group factors with like bases together.

B. (3mn2) (9m2n) (3mn2)(9m2n)

27m3n3

Multiply.


Multiply.

(6 3)(y3 y5)••

(3 9)(m m2)(n2 • n) ••

Multiply.

Example 1C: Multiplying Monomials

( ) ( ) − ÷ 2 2 21

124

s t st st

− 4 53s t


Multiply.

( )( )

− 22 2112

4ts tt s s

( )( ) −

2 2112

4t s ts ts

2

When multiplying powers with the same base, keep the base and add the exponents.

x2 • x3 = x2+3 = x5

Remember!

Check It Out! Example 1

Multiply.

a. (3x3)(6x2)

(3x3)(6x2)

(3 6)(x3 x2)••

18x5


Multiply.


Multiply.

b. (2r2t)(5t3)

(2r2t)(5t3) (2 5)(r2)(t3 t)••

10r2t4


Multiply.


Multiply.

c.

( )( )

4 52 2112

3x zy zx y

3

( )( )

2112

3x y x z y z

2 4 53

( )( )( )

3 22 4 5112 z

3zx x y y

7554x y z

To multiply a polynomial by a monomial, use the Distributive Property.

Multiply.

Example 2A: Multiplying a Polynomial by a Monomial

4(3x2 + 4x – 8)

4(3x2 + 4x – 8)

(4)3x2 +(4)4x – (4)8

12x2 + 16x – 32

Distribute 4.

Multiply.

6pq(2p – q)

(6pq)(2p – q)

Multiply.

Example 2B: Multiplying a Polynomial by a Monomial

(6pq)2p + (6pq)(–q)

(6 • 2)(p • p)(q) + (–1)(6)(p)(q • q)

12p2q – 6pq2

Distribute 6pq.

Group like bases together.

Multiply.

Multiply.Example 2C: Multiplying a Polynomial by a Monomial


Multiply.

( )+216

2x y xy x y8 2 2

x y( )+ 22 612

xyyx 8 2

x y x y( ) ( )

+2 216

28xy x y 22

12

x2 • x( )( ) ( )( ) +1• 6

2y • y x2 • x2 y • y2• 8

12

3x3y2 + 4x4y3

Distribute .21x y

2


Multiply.

a. 2(4x2 + x + 3)

2(4x2 + x + 3)

2(4x2) + 2(x) + 2(3)

8x2 + 2x + 6

Distribute 2.

Multiply.


Multiply.

b. 3ab(5a2 + b)

3ab(5a2 + b)

(3ab)(5a2) + (3ab)(b)

(3 • 5)(a • a2)(b) + (3)(a)(b • b)

15a3b + 3ab2

Distribute 3ab.


Multiply.


Multiply.c. 5r2s2(r – 3s)

5r2s2(r – 3s)

(5r2s2)(r) – (5r2s2)(3s)

(5)(r2 • r)(s2) – (5 • 3)(r2)(s2 • s)

5r3s2 – 15r2s3

Distribute 5r2s2.


Multiply.

To multiply a binomial by a binomial, you can apply the Distributive Property more than once:

(x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3.

Distribute x and 3 again.

Multiply.

Combine like terms.

= x(x + 2) + 3(x + 2)

= x(x) + x(2) + 3(x) + 3(2)

= x2 + 2x + 3x + 6

= x2 + 5x + 6

Another method for multiplying binomials is called the FOIL method.

4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6•

3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x•

2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x•

F

O

I

L

(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6

F O I L

1. Multiply the First terms. (x + 3)(x + 2) x x = x2•

Multiply.

Example 3A: Multiplying Binomials

(s + 4)(s – 2)

(s + 4)(s – 2)

s(s – 2) + 4(s – 2)

s(s) + s(–2) + 4(s) + 4(–2)

s2 – 2s + 4s – 8

s2 + 2s – 8

Distribute s and 4.

Distribute s and 4 again.

Multiply.

Combine like terms.

Multiply.

Example 3B: Multiplying Binomials

(x – 4)2

(x – 4)(x – 4)

(x x) + (x (–4)) + (–4 • x) + (–4 • (–4)) ••

x2 – 4x – 4x + 8

x2 – 8x + 8

Write as a product of two binomials.

Use the FOIL method.

Multiply.

Combine like terms.

Example 3C: Multiplying Binomials

Multiply.

(8m2 – n)(m2 – 3n)

8m2(m2) + 8m2(–3n) – n(m2) – n(–3n)

8m4 – 24m2n – m2n + 3n2

8m4 – 25m2n + 3n2


Multiply.

Combine like terms.

In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)

Helpful Hint

Check It Out! Example 3a

Multiply.

(a + 3)(a – 4)

(a + 3)(a – 4)

a(a – 4)+3(a – 4)

a(a) + a(–4) + 3(a) + 3(–4)

a2 – a – 12

a2 – 4a + 3a – 12

Distribute a and 3.

Distribute a and 3 again.

Multiply.

Combine like terms.

Check It Out! Example 3b

Multiply.

(x – 3)2

(x – 3)(x – 3)

(x x) + (x•(–3)) + (–3 • x)+ (–3)(–3) ●

x2 – 3x – 3x + 9

x2 – 6x + 9

Write as a product of two binomials.


Multiply.

Combine like terms.

Check It Out! Example 3c

Multiply.

(2a – b2)(a + 4b2)

(2a – b2)(a + 4b2)

2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)

2a2 + 8ab2 – ab2 – 4b4

2a2 + 7ab2 – 4b4


Multiply.

Combine like terms.

To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6):

(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)

= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)

= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)

= 10x3 + 50x2 – 30x + 6x2 + 30x – 18

= 10x3 + 56x2 – 18

You can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x2 + 10x – 6) and width (5x + 3):

2x2 +10x –6

10x3 50x2 –30x

30x6x2 –18

5x

+3

Write the product of the monomials in each row and column:

To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary. 10x3 + 6x2 + 50x2 + 30x – 30x – 18

10x3 + 56x2 – 18

Another method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers.

2x2 + 10x – 6

5x + 3×6x2 + 30x – 18

+ 10x3 + 50x2 – 30x 10x3 + 56x2 + 0x – 18

10x3 + 56x2 + – 18

Multiply each term in the top polynomial by 3.

Multiply each term in the top polynomial by 5x, and align like terms.

Combine like terms by adding vertically.

Simplify.

Multiply.

Example 4A: Multiplying Polynomials

(x – 5)(x2 + 4x – 6)

(x – 5 )(x2 + 4x – 6)

x(x2 + 4x – 6) – 5(x2 + 4x – 6)

x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)

x3 + 4x2 – 5x2 – 6x – 20x + 30

x3 – x2 – 26x + 30

Distribute x and –5.

Distribute x and −5 again.

Simplify.

Combine like terms.

Multiply.

Example 4B: Multiplying Polynomials

(2x – 5)(–4x2 – 10x + 3)

(2x – 5)(–4x2 – 10x + 3)

–4x2 – 10x + 32x – 5x

20x2 + 50x – 15+ –8x3 – 20x2 + 6x

–8x3 + 56x – 15

Multiply each term in the top polynomial by –5.

Multiply each term in the top polynomial by 2x, and align like terms.


Multiply.

Example 4C: Multiplying Polynomials

(x + 3)3

[(x + 3)(x + 3)](x + 3)

[x(x+3) + 3(x+3)](x + 3)

(x2 + 3x + 3x + 9)(x + 3)

(x2 + 6x + 9)(x + 3)

Write as the product of three binomials.

Use the FOIL method on the first two factors.

Multiply.

Combine like terms.

Example 4C: Multiplying Polynomials

Multiply.

(x + 3)3

x3 + 6x2 + 9x + 3x2 + 18x + 27

x3 + 9x2 + 27x + 27

x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9)

x(x2 + 6x + 9) + 3(x2 + 6x + 9)

Use the Commutative Property of Multiplication.

Distribute the x and 3.

Distribute the x and 3 again.

(x + 3)(x2 + 6x + 9)

Combine like terms.

Example 4D: Multiplying Polynomials

Multiply.(3x + 1)(x3 – 4x2 – 7)

x3 4x2 –7

3x4 12x3 –21x

4x2x3 –7

3x

+1

3x4 + 12x3 + x3 + 4x2 – 21x – 7

Write the product of the monomials in each row and column.

Add all terms inside the rectangle.

3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms.

A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or 6 terms before simplifying.

Helpful Hint

•

Check It Out! Example 4a

Multiply.

(x + 3)(x2 – 4x + 6)

(x + 3 )(x2 – 4x + 6)

x(x2 – 4x + 6) + 3(x2 – 4x + 6)

Distribute x and 3.

Distribute x and 3 again.

x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6)

x3 – 4x2 + 3x2 +6x – 12x + 18

x3 – x2 – 6x + 18

Simplify.

Combine like terms.

Check It Out! Example 4b

Multiply.

(3x + 2)(x2 – 2x + 5)

(3x + 2)(x2 – 2x + 5)

x2 – 2x + 5 3x + 2×

Multiply each term in the top polynomial by 2.

Multiply each term in the top polynomial by 3x, and align like terms.2x2 – 4x + 10

+ 3x3 – 6x2 + 15x3x3 – 4x2 + 11x + 10


Example 5: ApplicationThe width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.a. Write a polynomial that represents the area of the

base of the prism.

Write the formula for the area of a rectangle.

Substitute h – 3 for w and h + 4 for l.

A = l • w

A = l w•

A = (h + 4)(h – 3)

Multiply.A = h2 + 4h – 3h – 12

Combine like terms.A = h2 + h – 12

The area is represented by h2 + h – 12.

Example 5: ApplicationThe width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.b. Find the area of the base when the height is 5 ft.

A = h2 + h – 12

A = h2 + h – 12

A = 52 + 5 – 12

A = 25 + 5 – 12

A = 18

Write the formula for the area the base of the prism.

Substitute 5 for h.

Simplify.

Combine terms.

The area is 18 square feet.


The length of a rectangle is 4 meters shorter than its width.

a. Write a polynomial that represents the area of the rectangle.

Write the formula for the area of a rectangle.

Substitute x – 4 for l and x for w.

A = l w•

A = l w•

A = x(x – 4)

Multiply.A = x2 – 4x

The area is represented by x2 – 4x.

Check It Out! Example 5 The length of a rectangle is 4 meters shorter than its width.

b. Find the area of a rectangle when the width is 6 meters. A = x2 – 4x

A = x2 – 4x

A = 36 – 24

A = 12

Write the formula for the area of a rectangle whose length is 4 meters shorter than width .

Substitute 6 for x.

Simplify.

Combine terms.

The area is 12 square meters.

A = 62 – 4 • 6

Lesson Quiz: Part I

Multiply.

1. (6s2t2)(3st)

2. 4xy2(x + y)

3. (x + 2)(x – 8)

4. (2x – 7)(x2 + 3x – 4)

5. 6mn(m2 + 10mn – 2)

6. (2x – 5y)(3x + y)

4x2y2 + 4xy3

18s3t3

x2 – 6x – 16

2x3 – x2 – 29x + 28

6m3n + 60m2n2 – 12mn

6x2 – 13xy – 5y2

Lesson Quiz: Part II

7. A triangle has a base that is 4cm longer than its height.a. Write a polynomial that represents the area

of the triangle.

b. Find the area when the height is 8 cm.

48 cm2

12

h2 + 2h

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Algebra unit 8.2 8.3