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Multiplying

Polynomials

I will multiply polynomial expressions using the

distributive property and exponent rules

Multiply.

Example 1: Multiplying Monomials

A. (6y3)(3y5)

(6y3)(3y5)

18y8

Group factors with like bases

together.

B. (3mn2) (9m2n)

(3mn2)(9m2n)

27m3n3

Multiply.

Group factors with like bases

together.

Multiply.

(6 3)(y3 y5)

(3 9)(m m2)(n2 n)

Multiply.

Example 1C: Multiplying Monomials

2 2 2112

4s t st st 4 53s t

Group factors with like

bases together.

Multiply.

22 2112

4ts tt s s

g gg g g

2 2112

4ts 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

Group factors with like bases

together.

Multiply.

Group factors with like bases

together.

Multiply.

b. (2r2t)(5t3)

(2r2t)(5t3)

(2 5)(r2)(t3 t)

10r2t4

Check It Out! Example 1

Multiply.

Group factors with

like bases

together.

Multiply.

c.

4 52 2112

3x zy zx y

3

2112

3x y x z y z

2 4 53

g gg g

3 22 4 5112 z

3zx x y y

7554x y z

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

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.

Check It Out! Example 2

Multiply.

a. 2(4x2 + x + 3)

2(4x2 + x + 3)

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

8x2 + 2x + 6

Distribute 2.

Multiply.

Check It Out! Example 2

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.

Group like bases

together.

Multiply.

Check It Out! Example 2

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.

Group like bases

together.

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

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

Use the FOIL method.

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.

Use the FOIL method.

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

Use the FOIL method.

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

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.

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 + 53x + 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

3x3 – 4x2 + 11x + 10Combine like terms by

adding vertically.

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

1

2h2 + 2h