ThenThenYou have already solved equations by dividing. (Lesson 4-4)

NowNow Divide polynomials by

monomials.

Solve problems using division of polynomials.

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6Looking

Ahead

Lesson 6 Dividing a Polynomial by a Monomial LA21

Why?Why?

Student Council is selling milkshakes at lunch

as a fundraiser. Each milkshake requires 1 _ 8 gallon of ice

cream. They had 6 1 _ 2 gallons of ice cream. Then

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.

Dividing Polynomials

Words To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Symbols a + b _ c = a _ c + b _ c

Key Concept

EXAMPLE 1 Divide a Polynomial by a Monomial

Divide.

a. (9 b 2 - 15b) ÷ 3b

(9 b 2 - 15b) ÷ 3b = 9 b 2 - 15b _ 3b

Write as a rational expression.

= 9 b 2 _

3b - 15b _

3b Divide each term by 3b.

= 9 _ 3 · b 2

_ b - 15

_ 3 · b _

b Associative Property

= 3 · b 2 - 1 - 5 · 1 Quotient of Powers and Identity Properties

= 3b - 5 Simplify.

b. (6 x 2 + 4x) ÷ 2x

(6 x 2 + 4x) ÷ 2x = 6 x 2 + 4x _

2x Write as a rational expression.

= 6 x 2 _

2x + 4x _

2x Divide each term by 2x.

= 6 _ 2 · x 2

_ x + 4 _ 2 · x _ x Associative Property

= 3 · x 2 - 1 + 2 · 1 Quotient of Powers and Identity Properties

= 3x + 2 Simplify.

Guided Practice✓ 1A. (10 x 2 y 2 + 5xy) ÷ 5xy 1B. (27 x 2 - 21 y 2 ) ÷ 3

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Dividing a Polynomial by a Monomial

LA22 Looking Ahead to Algebra 1

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) ÷ 6 = 6x + 12

_ 6 Write as a rational expression.

= 6x _ 6 + 12

_ 6 Divide each term by 6.

= x + 2 Simplify.

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 8 x 2 + 12x. If the width of the garden is 4x, what is the length of the garden?

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8x2 12x8x2 12x

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

✓ Check Your Understanding

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. (18 x 2 + 32x) ÷ 2x 8. (20 k 2 - 35k) ÷ 5k

9. (20 b 3 + 40b) ÷ 20b 10. (4 x 3 + 2 x 2 - 6x) ÷ 2x

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?

Example 1p. LA21

Example 1p. LA21

Example 2p. LA22

Example 2p. LA22

12. GEOMETRY The perimeter of the square below can be shown by the expression 12 x 2 + 24x. What is the length of each side of the square?

= 12x2+ 24x= 12x2+ 24x

12. GEOMETRY The perimeter of the square below can be shown by the expression 12 x 2 + 24x. What is the length of each side of the square?

= 12x2+ 24x= 12x2+ 24x

A = 16x + 4yA = 16x + 4y

Lesson 6 Dividing a Polynomial by a Monomial LA23

Practice and Problem Solving

Divide.

13. (16x + 4 y 2 ) ÷ 4 14. (15a + 3 b 2 ) ÷ 3

15. (21 a 2 b - 14 a 2 ) ÷ 7 a 2 16. (36 st 2 - 9st) ÷ 9st

17. (13 r 2 s - 26 r 2 ) ÷ 13 r 2 18. (32np + m 2 n p 2 ) ÷ np

19. (42 c 3 d 2 + 56 c 2 d - 14c) ÷ 7c 20. (81 m 3 n 2 - 45 m 2 n - 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 3 x 3 + 6 x 2 - 30x. If the height of the basement is 3x, what is the area of its base?

22. BASKETBALL The length of a basketball court is 6 x 2 + 20x. If the length is 2x times the width, what is the width of the court?

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

(15 x 3 + 12 x 2 - 24x) ÷ 3x. Is either of them correct? Explain your reasoning.

George

= 15 x 3 + 12 x 2 – 24x __

3x

= 15 x 3 _ 3x

+ 12 x 2 _ 3x

– 24x _

3x

= 5 x 2 + 4x – 8

Marilyn

= 15 x 3 + 12 x 2 – 24x _

3x

= 15 x 3 _

3x + 12 x 2

_ 3x

– 24x _

3x

= 5 x 3 + 4 x 2 – 8x

26. REASONING Explain why dividing a polynomial by a monomial could be called factoring.

27. WRITING IN MATH Explain why 2 y 2 + 3 cannot be divided by 2y without using fractions.

Example 1p. LA21

Example 1p. LA21

Example 2p. LA22

Example 2p. LA22

= 3x3+ 6x2

- 30x

h = 3x

= 3x3+ 6x2

- 30x

h = 3x

6x 2+ 20x

w = ?

6x 2+ 20x

w = ?