7-8 multiplying polynomials to multiply monomials and polynomials, you will use some of the...

7-8 Multiplying Polynomials

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


Multiply.

Additional 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.

Additional Example 1C: Multiplying Monomials

2 2 21 124 s t st st

4 53s t


Multiply.

22 21 124 ts tt s s

2 21 124 t s ts ts

2


When multiplying powers with the same base, keep the base and add the exponents. x2 x3 = x2+3 = x5

Remember!

7-8 Multiplying PolynomialsCheck 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 21 123 x zy zx y

3

1 (12x3z2)(y4z5)3 x2y

3 22 4 51 12 z3 zx x y y7554x y z


Check up: p. 451 #’s 3,4,6


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


Multiply.

Additional 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.

Additional 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.

Additional Example 2C: Multiplying a Polynomial by a Monomial


Multiply.

12 x2y (6xy + 8x2y2)

x y 22 612 xyyx 8 2

x y x y

2 21 62 8xy x y 22

12

x2 • x

1

• 62 y • y x2 • x2 y • y2• 8

12

3x3y2 + 4x4y3

Distribute .21 x y2


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.


Check up: p. 451 #’s 10,12


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

7-8 Multiplying PolynomialsAnother 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.

Additional 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 – 8s2 + 2s – 8

Distribute s and 4.

Distribute s and 4 again.

Multiply.

Combine like terms.


Multiply.

Additional Example 3B: Multiplying Binomials

(x – 4)2

(x – 4)(x – 4)

x(x) + x(–4) – 4(x) – 4(–4)

x2 – 4x – 4x + 16

x2 – 8x + 16

Write as a product of two binomials.

Use the FOIL method.

Multiply.

Combine like terms.

7-8 Multiplying PolynomialsAdditional 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

7-8 Multiplying PolynomialsCheck 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.

7-8 Multiplying PolynomialsCheck 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.

7-8 Multiplying PolynomialsCheck It Out! Example 3c

Multiply.(2a – b2)(a + 4b2)

2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)

2a2 + 8ab2 – ab2 – 4b4

2a2 + 7ab2 – 4b4


Multiply.

Combine like terms.


Check up: p. 451 #’s 14, 17

7-8 Multiplying PolynomialsTo 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

7-8 Multiplying PolynomialsYou 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 –610x3 50x2 –30x

30x6x2 –185x

+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

7-8 Multiplying PolynomialsAnother 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 + 36x2 + 30x – 18

Multiply each term in the top polynomial by 3.

Multiply each term in the top polynomial by 5x, and align like terms.+ 10x3 + 50x2 – 30x

10x3 + 56x2 + 0x – 18

10x3 + 56x2 – 18

Combine like terms by adding vertically.

Simplify.


Multiply.

Additional 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 – 6x – 5x2 – 20x + 30

x3 – x2 – 26x + 30

Distribute x and –5.

Distribute x and −5 again.

Simplify.

Combine like terms.


Multiply.

Additional Example 4B: Multiplying Polynomials

(2x – 5)(–4x2 – 10x + 3)

(2x – 5)(–4x2 – 10x + 3)

–4x2 – 10x + 32x – 5×

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.

Additional Example 4C: Multiplying Polynomials

(x + 3)3

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

[x(x) + x(3) + 3(x) +3(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.

7-8 Multiplying PolynomialsAdditional Example 4C Continued

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.

7-8 Multiplying PolynomialsAdditional Example 4D: Multiplying Polynomials

Multiply.(3x + 1)(x3 + 4x2 – 7)

x3 4x2 –73x4 12x3 –21x

4x2x3 –73x+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

7-8 Multiplying PolynomialsCheck 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 + 6x + 3x2 – 12x + 18

x3 – x2 – 6x + 18

Simplify.

Combine like terms.

7-8 Multiplying PolynomialsCheck 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



Check up: p. 451 #’s 22,24

7-8 Multiplying PolynomialsAdditional Example 5: Application

The 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 – 3h + 4h – 12Combine like terms.A = h2 + h – 12

The area is represented by h2 + h – 12.

7-8 Multiplying PolynomialsAdditional Example 5: Application

The 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 = 52 + 5 – 12A = 25 + 5 – 12A = 18

Use the polynomial from part A.

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 – 4)xMultiply.A = x2 – 4x

The area is represented by x2 – 4x.


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 = 36 – 24A = 12

Use the polynomial from part a.

Substitute 6 for x.

Simplify.

Combine terms.

The area is 12 square meters.

A = 62 – 4(6)


Check up: p. 451 # 25

7-8 multiplying polynomials to multiply monomials and polynomials, you will use some of the...

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