Notes 7-5
Multiplying a Monomial by
a Polynomial
I. Multiplying Monomials by Polynomials
To multiply monomials and polynomials,
you will use some of the properties of
exponents that you learned earlier in this
chapter.
Remember!
When multiplying powers with the same base, keep the base
and add the exponents.
x2  x3 = x2+3 = x5
To multiply a polynomial by a monomial,
use the Distributive Property.
A. Multiply
4(3x2 + 4x – 8)
4(3x2 + 4x – 8)
Distribute 4.
(4)3x2 + (4)4x – (4)8
Multiply.
12x2 + 16x – 32
B. Multiply.
3x(2x2 - 5x + 7)
3x(2x2 - 5x + 7)
Distribute 3x.
3x(2x2) + 3x(-5x) + 3x(7)
Multiply.
6x3 - 15x2 + 21x
C. Multiply.
6pq(2p – q)
(6pq)(2p – q)
Distribute 6pq.
(6pq)2p + (6pq)(–q)
12p2q – 6pq2
Multiply.
D. Multiply.
2x2 (6x2 + 7x - 12)
2x2 (6x2 + 7x - 12)
Distribute 2x2.
2x2(6x2) + 2x2(7x) + 2x2(-12)
Multiply.
12x4 + 14x3 - 24x2
On your whiteboards
E. Multiply.
b. 3ab(5a2 + b)
3ab(5a2 + b)
Distribute 3ab.
(3ab)(5a2) + (3ab)(b)
Group like bases together.
(3  5)(a  a2)(b) + (3)(a)(b  b)
15a3b + 3ab2
Multiply.
On your whiteboards
F. Multiply.
c. 5r2s2(r – 3s)
5r2s2(r – 3s)
Distribute 5r2s2.
(5r2s2)(r) – (5r2s2)(3s)
5r3s2 – 15r2s3
Multiply.
III. Solving Equations Involving Polynomials
Ex 1:
5x  3( x  4)  28 Write the original equation.
5x  3x  12  28 Distribute the 3.
8x  12  28 Combine like terms.
12  12 Subtract from both sides.
8x  16
8
8
x2
Simplify
Divide both sides.
Simplify.
CHECK
Example 2:
4 x  3( x  2)  21 Write the original equation.
Distribute the 3 and the
4 x  3x  6  21
negative.
x  6  21 Combine like terms.
 6  6
x  15
Subtract
Simplify from both sides.
CHECK
Ex 3:
c2 + 3c - c2 + 4c = 9c - 16
Distribute both c’s on left
7c = 9c - 16
Simplify: positive and
negative c2’s cancel each
other out, 3c + 4c = 7c
2c = - 16
c=-8
Subtract 9c from both
sides
Divide both sides by 2
III. Applications
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 (x + 4)(6x)
A = (½)[6x(x) + 6x(4)]
A = (½)(6x2 + 24x)
A = (½)(6x2)+ (½)(24x)
A = 3x2+ 12x
6x
X+4
Write an expression that represents the area of
the shaded region in terms of x.
x+2
2)
3)
3
9
6
55
xx ++ 22
2x + 5
3x + 7
6(2x  5)  3(x  2)
9(3x  7)  5(x  2)
12x  30  3x  6
27x  63  5x  10
15x  36
22x 53
Write an expression that represents the area of
the shaded region in terms of x.
2
2
2x
4
3x

2x
4)
5)
5
xx2288
8
7
33
6x2  5x
7(6x  5x)  5(3x  2x)
8(2x  4)  3(x 2  8)
42x  35x  15x 10x
16x  32  3x  24
2
2
2
2
57x  25x
2
2
2
2
13x  56
2
6. Write a polynomial to represent the shaded region.
Shaded Region:
Big rectangle – small rectangle
lbwb - lsws
2x
3x - 2
(3x2 + 6x – 1)(3x) – (3x - 2)(2x)
9x3 + 18x2 - 3x – (6x2 - 4x)
9x3 + 18x2 - 3x - 6x2 + 4x
9x3 + 12x2 + x
3x2 + 6x - 1
3x
Using Polynomials in Real Life
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.
Write a model for the area of the mat around the photograph as a function of the
scale factor.
Use a verbal model.
Verbal Model
Area of mat = Total Area –
Labels
…
Area of
photo
7x
Area of mat = A
(square inches)
5x
Total Area = (10x)(14x – 2)
(square inches)
10x
Area of photo = (5x)(7x)
(square inches)
14x – 2
SOLUTION
Using Polynomials in Real Life
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.
Write a model for the area of the mat around the photograph as a function of the
scale factor.
SOLUTION
…
= 140x 2 – 20x – 35x 2
5x
= 105x 2 – 20x
10x
14x – 2
Algebraic
Model
7x
A = (10x)(14x – 2) – (5x)(7x)
A model for the area of the mat around the photograph as a function of the
scale factor x is A = 105x 2 – 20x.