Notes 7-5

Multiplying a Monomial by a Polynomial

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

x2 x3 = x2+3 = x5

Remember!

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

I. Multiplying Monomials by Polynomials

A. Multiply

4(3x2 + 4x – 8)

4(3x2 + 4x – 8)

(4)3x2 + (4)4x – (4)8

12x2 + 16x – 32

Distribute 4.

Multiply.

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

B. Multiply.

3x(2x2 - 5x + 7)

3x(2x2 - 5x + 7)

3x(2x2) + 3x(-5x) + 3x(7)

6x3 - 15x2 + 21x

Distribute 3x.

Multiply.

6pq(2p – q)

(6pq)(2p – q)

C. Multiply.

(6pq)2p + (6pq)(–q)

12p2q – 6pq2

Distribute 6pq.

Multiply.

D. Multiply.

2x2 (6x2 + 7x - 12)

2x2 (6x2 + 7x - 12)

2x2(6x2) + 2x2(7x) + 2x2(-12)

12x4 + 14x3 - 24x2

Distribute 2x2.

Multiply.

E. 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.

On your whiteboards

F. Multiply.

c. 5r2s2(r – 3s)

5r2s2(r – 3s)

(5r2s2)(r) – (5r2s2)(3s)

5r3s2 – 15r2s3

Distribute 5r2s2.

Multiply.

On your whiteboards

Ex 1:

28)4(35 xx Write the original equation.

281235 xx Distribute the 3.

28128 x Combine like terms.

Subtract from both sides.

2x

Divide both sides.

CHECK

1212

Simplify 168 x

8 88

Simplify.

III. Solving Equations Involving Polynomials

21)2(34 xx Write the original equation.

21634 xxDistribute the 3 and the negative.

216 x Combine like terms.

Subtract from both sides. 15x

CHECK

66

Simplify

Example 2:

Ex 3:

c2 + 3c - c2 + 4c = 9c - 16

7c = 9c - 16

Distribute both c’s on left

Simplify: positive and

negative c2’s cancel each

other out, 3c + 4c = 7c

Subtract 9c from both

sides 2c = - 16

Divide both sides by 2 c = - 8

1. A triangle has a base of (x + 4) and a height of 6x. Find the area of the rectangle.

A = (½)bh

A = (½)[6x(x) + 6x(4)] X + 4

6x A = ½ (x + 4)(6x)

A = (½)(6x2 + 24x)

A = (½)(6x2)+ (½)(24x)

A = 3x2+ 12x

III. Applications

Write an expression that represents the area of the shaded region in terms of x.

2) 3) 3

6

2x + 5

x + 2

6(2x 5) 3(x 2)

12x 30 3x 6

15x 36

9

3x + 7

9(3x 7) 5(x 2)

27x 63 5x 10

22x

x + 2

53

5 5

x + 2

Write an expression that represents the area of the shaded region in terms of x.

4) 5) 5

7

27(6x 5x) 2 5(3x 2x)

242x 35x 2 15x 10x

257x 25x

8

28(2x 4) 2 3(x 8)

216x 32 2 3x 24

213x 56

3

23x 2x

26x 5x

22x 4

2x 8

3

2x 8

6. Write a polynomial to represent the shaded region.

3x - 2

2x

3x2 + 6x - 1

3x

Shaded Region: Big rectangle – small rectangle

lbwb - lsws

(3x2 + 6x – 1)(3x) – (3x - 2)(2x)

9x3 + 18x2 - 3x – (6x2 - 4x)

9x3 + 18x2 - 3x - 6x2 + 4x

9x3 + 12x2 + x

Total Area = (10x)(14x – 2) (square inches)

Area of photo =

You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on

a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches

less than twice as high as the enlarged photo.

Using Polynomials in Real Life

Write a model for the area of the mat around the photograph as a function of the

scale factor.

Verbal Model

Labels

Area of mat = Area of

photo

Area of mat = A

(5x)(7x)

(square inches)

(square inches)

Total Area –

Use a verbal model.

5x

7x

14x –

2

10x

SOLUTION

…

(10x)(14x – 2) – (5x)(7x)

You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on

a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches

less than twice as high as the enlarged photo.

Using Polynomials in Real Life

Write a model for the area of the mat around the photograph as a function of the

scale factor.

A =

= 140x 2 – 20x – 35x 2

SOLUTION

= 105x 2 – 20x

A model for the area of the mat around the photograph as a function of the

scale factor x is A = 105x 2 – 20x.

Algebraic

Model

…

5x

7x

14x –

2

10x