Looking
6
Ahead
Dividing a Polynomial by a Monomial
Why?
Then
You have already solved
equations by dividing.
(Lesson 4-4)
Now
Divide polynomials by
monomials.
Solve problems using
division of
polynomials.
Student Council is selling milkshakes at lunch
1
as a fundraiser. Each milkshake requires _
gallon of ice
8
1
_
cream. They had 6 gallons of ice cream. Then
2
their advisor brought them 5 more gallons.
a. Find the number of milkshakes that can be sold with
the amount of ice cream they now have.
b. Describe two ways to find the number of milkshakes.
Dividing Polynomials By Monomials To divide a polynomial by a monomial,
divide each term of the polynomial by the monomial.
Math Online
glencoe.com
Key Concept
Words
Symbols
Dividing Polynomials
To divide a polynomial by a monomial, divide each term
of the polynomial by the monomial.
a+b
a
b
_
=_+_
EXAMPLE 1
c
c
c
Divide a Polynomial by a Monomial
Divide.
a. (9b 2 - 15b) ÷ 3b
2
9b - 15b
(9b 2 - 15b) ÷ 3b = _
3b
Write as a rational expression.
2
9b
15b
=_
-_
3b
3b
2
9
15 _
b
=_·_-_
·b
3 b
3 b
Divide each term by 3b.
Associative Property
= 3 · b2 - 1 - 5 · 1
Quotient of Powers and Identity Properties
= 3b - 5
Simplify.
b. (6x 2 + 4x) ÷ 2x
6x 2 + 4x
2x
2
6
x
4x
=_+_
2x
2x
2
6
x
4 _
=_·_
+_
·x
2 x
2 x
(6x 2 + 4x) ÷ 2x = _
✓Guided Practice
Write as a rational expression.
Divide each term by 2x.
Associative Property
= 3 · x2 - 1 + 2 · 1
Quotient of Powers and Identity Properties
= 3x + 2
Simplify.
1A. (10x 2y 2 + 5xy) ÷ 5xy
1B. (27x 2 - 21y 2) ÷ 3
Personal Tutor glencoe.com
Lesson 6 Dividing a Polynomial by a Monomial
LA21
Solve Problems You can use division to solve real-world problems.
EXAMPLE 2
Solve Problems
ZOOS Six friends visited the zoo to see the new panda exhibit. The group
paid for admission and an additional $12 for parking. The total cost of the
visit can be shown by the expression $6x + $12. What was the cost of the
visit for one person?
6x + 12
(6x + 12) ÷ 6 = _
6
There are about 110
pandas in captivity. Only
seven of these are in the
United States. The majority
of the other pandas in
captivity are found in
China.
Source: Panda Bear Facts
Write as a rational expression.
6x
12
=_
+_
Divide each term by 6.
=x+2
Simplify.
6
6
So, each person paid x + 2 dollars for admission to the zoo.
✓Guided Practice
2. GARDENS The model at the right represents a garden.
The total area of the garden shown can be represented
by the expression 8x 2 + 12x. If the width of the garden
is 4x, what is the length of the garden?
8x 2
12x
Personal Tutor glencoe.com
✓ Check Your Understanding
Example 1
p. LA21
Divide.
1. (5abc + c) ÷ c
2. (14ab + 28b) ÷ 14b
3. (16x + 24xy) ÷ 8x
4. (25st - 35s) ÷ 5s
5. (30mn - 9m) ÷ 3m
6. (42q + 56) ÷ 7
7. (18x 2 + 32x) ÷ 2x
8. (20k 2 - 35k) ÷ 5k
9. (20b 3 + 40b) ÷ 20b
Example 2
p. LA22
11. GEOMETRY The area of the
rectangle below can be shown
by the expression (16x + 4y)
square feet. If the width of the
rectangle is 4 feet, what is the
length of the rectangle?
10. (4x 3 + 2x 2 - 6x) ÷ 2x
12. GEOMETRY The perimeter of the
square below can be shown by the
expression 12x 2 + 24x. What is the
length of each side of the square?
A = 16x + 4y
1= 12x 2 + 24x
LA22 Looking Ahead to Algebra 1
Practice and Problem Solving
Example 1
p. LA21
Example 2
p. LA22
Divide.
13. (16x + 4y 2) ÷ 4
14. (15a + 3b 2) ÷ 3
15. (21a 2b - 14a 2) ÷ 7a 2
16. (36st 2 - 9st) ÷ 9st
17. (13r 2s - 26r 2) ÷ 13r 2
18. (32np + m 2np 2) ÷ np
19. (42c 3d 2 + 56c 2d - 14c) ÷ 7c
20. (81m 3n 2 - 45m 2n - 27n) ÷ 9n
21. CONSTRUCTION A contractor built
a basement for a new home. The
volume of the rectangular basement
shown at the right is represented by the
expression 3x 3 + 6x 2 - 30x. If the
height of the basement is 3x,
what is the area of its base?
h = 3x
7= 3x 3 +
6x 2
- 30x
22. BASKETBALL The length of a
basketball court is 6x 2 + 20x.
If the length is 2x times the width,
what is the width of the court?
w=?
2
6x + 20x
H.O.T. Problems
Use Higher-Order Thinking Skills
23. OPEN ENDED Write a polynomial and a monomial that have a quotient of
x 2 + 3x + 5.
24. CHALLENGE The length and width of a rectangle are represented by 2x and
9 - 4x. If x must be an integer, what are the only possible measures for the
area of this rectangle?
25. ERROR ANALYSIS George and Marilyn are finding the quotient of
(15x 3 + 12x 2 - 24x) ÷ 3x. Is either of them correct? Explain your reasoning.
George
3
Marilyn
2
x + 12x – 24x
__
= 15
3x
24x
15x 3 + _
12x 2 – _
=_
3x
2
3x
= 5x + 4x – 8
3x
3
2
15x + 12x – 24x
=_
3x
3
15
x
12x 2 24x
=_+_–_
3x
3x
3x
= 5x 3 + 4x 2 – 8x
26. REASONING Explain why dividing a polynomial by a monomial could be
called factoring.
27. WRITING IN MATH Explain why 2y 2 + 3 cannot be divided by 2y without
using fractions.
Lesson 6 Dividing a Polynomial by a Monomial
LA23