Multiplying Polynomials

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Multiply polynomials.

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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)

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Student Example: Example 1

Multiply.

a. (3x3)(6x2) (3x3)(6x2)

(3 6)(_ x3 _x2) 18x5

Multiply.

Multiply. b. (2r2t)(5t3)

(2r2t)(5t3)

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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.

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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.

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Student Example: 2a

Multiply.

a. 2(4x2 + x + 3)

2(4x2 + x + 3)

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

8x2 + 2x + 6

Distribute 2.

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Student Example: 2b

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.

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Student Example: 2c

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.

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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 _{+ 2}_x_{+ 3}_x_{+ 6}

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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₊₃_x₊ ₆₌_x2_{+ 5}_x_{+ 6}

F O I L

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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.

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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.

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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.

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

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Student 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.

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Student 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.

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Student 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.

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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

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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.

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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.

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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)

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Student 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.

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Student 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 + 15x

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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 + 4h – 3h – 12

Combine like terms. A = h2 + h – 12

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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 = 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.

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Student Example: 5

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

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Student Example 5 continued

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.

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Exit Ticket

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