Download Multiplying and Dividing Rational Expressions

LESSON
9.2
Name
Multiplying and
Dividing Rational
Expressions
Class
9.2
Date
Multiplying and Dividing
Rational Expressions
Essential Question: How can you multiply and divide rational expressions?
Resource
Locker
Common Core Math Standards
The student is expected to:
Explore
A-APR.7(+)
Use the facts you know about multiplying rational numbers to determine how to multiply rational expressions.
Understand that rational expressions form a system analogous to the
rational numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply, and
divide rational expressions. Also F-BF.1b, A-CED.4
A
Mathematical Practices
Language Objective
20
5 =_
4 ⋅_
_
5 6
30
C
To simplify, factor the numerator and denominator.
20 = 2 ⋅ 2 ⋅ 5
Explain to a partner the steps for multiplying and dividing rational
expressions.
30 = 2 ⋅ 3 ⋅ 5
D
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
Cancel common factors in the numerator and denominator to simplify the product.
2
2⋅2⋅5 = _
5 =_
20 = _
4 ⋅_
_
5 6
30
2⋅3⋅5
3
E
Based on the steps used for multiplying rational numbers, how can you multiply the rational
x + 1 ______
expression ____
⋅ 3 ?
x-1
2(x + 1)
Multiply x + 1 by 3 to find the numerator of the product, and multiply x - 1 by 2(x + 1) to
find the denominator. Then cancel common factors to simplify the product.
Reflect
1.
Discussion Multiplying rational expressions is similar to multiplying rational numbers. Likewise, dividing
rational expressions is similar to dividing rational numbers. How could you use the steps for dividing
rational numbers to divide rational expressions?
When dividing rational numbers, multiply by the reciprocal of the divisor and follow the
steps for multiplying rational numbers. So, when dividing rational expressions, multiply
by the reciprocal of the divisor and follow the steps for multiplying rational expressions.
Module 9
PREVIEW: LESSON
PERFORMANCE TASK
Lesson 2
439
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
HARDCOVER PAGES 317324
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A2_MNLESE385894_U4M09L2.indd 439
Resource
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Harcour t
To find the product of rational expressions, factor
each numerator and denominator, multiply the
numerators and denominators, and simplify the
resulting rational expression that is the product. To
find the quotient of rational expressions, multiply
the dividend by the reciprocal of the divisor and
then follow the steps for multiplying rational
expressions.
5?
4 ⋅_
How do you multiply _
5 6
Multiply 4 by 5 to find the numerator of the product, and multiply 5 by 6 to find
the denominator .
B
MP.8 Patterns
Essential Question: How can you
multiply and divide rational
expressions?
Relating Multiplication Concepts
439
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Lesson 9.2
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05/04/14
11:22 AM
05/04/14 11:22 AM
Explain 1
Multiplying Rational Expressions
EXPLORE
To multiply rational expressions, multiply the numerators to find the numerator of the product, and multiply the
denominators to find the denominator. Then, simplify the product by cancelling common factors.
Relating Multiplication Concepts
Note the excluded values of the product, which are any values of the variable for which the expression is undefined.
Example 1

Find the products and any excluded values.
2x 2-6x-20
3x 2
_
⋅ __
x 2-2x-8 x 2-3x-10
INTEGRATE TECHNOLOGY
2(x + 2)(x - 5)
3x
3x
2x -6x-20 = __
_
⋅ __
⋅ __
x 2-2x-8 x 2-3x-10
(x + 2)(x - 4) (x + 2)(x - 5)
2
2
2
6x 2(x + 2)(x - 5)
= ___
(x + 2)(x - 4)(x + 2)(x - 5)
6x 2(x + 2)(x - 5)
= ___
(x + 2)(x - 4)(x + 2)(x - 5)
Students have the option of completing the Explore
activity either in the book or online.
Factor the numerators and
denominators.
Multiply the numerators and
multiply the denominators.
QUESTIONING STRATEGIES
Cancel the common factors in the
numerator and denominator.
What are two different ways of
2x2 y 3y
multiplying _____ · ___ ? Multiply across and
6xy 4x
then simplify the result, or divide out common
factors of the numerators and denominators and
then multiply across. In either case, the result
y
will be __.
4
6x 2
= __
(x + 2)(x - 4)
Determine what values of x make each expression undefined.
3x 2
__
:
The denominator is 0 when x = -2 and x = 4.
x 2 - 2x - 8
2
2x - 6x - 20 : The denominator is 0 when x = -2 and x = 5.
__
x 2 - 3x - 10
Excluded values: x = -2, x = 4, and x = 5

7x + 35
x 2 - 8x
__
⋅_
x+8
14(x 2 + 8x + 15)
(
)
x (x - 8)
7 x+5
7x + 35
x 2 - 8x
__
⋅ _ = __ ⋅ __ Factor the numerators and
x+8
x+8
14(x 2 + 8x + 15)
denominators.
14 x + 3 (x + 5)
(
x(x - 8)
= __
2(x + 3)(x + 8)
Multiply the numerators and
multiply the denominators.
Cancel the common factors in the
numerator and denominator.
Determine what values of x make each expression undefined.
x 2 - 8x
__
: The denominator is 0 when x = -3 and x = -5 .
14(x 2 + 8x + 15)
7x + 35
_
:
The denominator is 0 when x = -8 .
x+8
Excluded values:
EXPLAIN 1
Multiplying Rational Expressions
© Houghton Mifflin Harcourt Publishing Company
)
7x(x - 8)( x + 5 )
= ___
14( x + 3 )(x + 5)(x + 8)
AVOID COMMON ERRORS
Students sometimes confuse multiplying rational
expressions with cross-multiplying. Point out that
cross-multiplying takes place across an equal sign
c . Tell
a = __
when solving equations of the form __
b d
students to use the equal sign as the cue to cross
multiply. When multiplying rational expressions,
multiply straight across.
x = -3, x = -5, and x = -8
Module 9
440
QUESTIONING STRATEGIES
Lesson 2
PROFESSIONAL DEVELOPMENT
A2_MNLESE385894_U4M09L2.indd 440
Learning Progressions
Students learned how to simplify rational expressions in the previous lesson. They
also know how to multiply and divide numerical fractions. Here, they combine
those skills to multiply and divide rational expressions. Students apply their
knowledge of factoring, as well as of multiplying polynomials, to simplify
expressions involving multiplication and division of rational expressions. The
concept of excluded values will carry over into later studies, for example, in
excluding extraneous values in the simplification of logarithms.
16/10/14 2:09 PM
Why should you factor the numerators and
the denominators before you multiply? It
makes it easier to multiply because you can divide
out common factors from a numerator and a
denominator before multiplying.
Multiplying and Dividing Rational Expressions
440
Your Turn
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Students should recognize that multiplying
Find the products and any excluded values.
2.
x2 - 9
x-8
__
⋅_
x 2 - 5x - 24 2x 2 - 18x
=
two rational expressions does not introduce excluded
values. The excluded values of the product are the
combined excluded values of the original rational
expressions. Students can use this fact to help detect
errors in their work.
=
(x + 3)(x - 8)
2x(x - 9)
(x + 3)(x - 3)(x - 8)
___________________
2x(x + 3)(x - 8)(x - 9)
3x
=_
x+1
(x - 3)
2x(x - 9)
Excluded values: x = -1 and x = 9
Excluded values: x = -3, x = 8, x = 0, and x = 9
Explain 2
Dividing Rational Expressions
To divide rational expressions, change the division problem to a multiplication problem by multiplying by the
reciprocal. Then, follow the steps for multiplying rational expressions.
Dividing Rational Expressions
Example 2
Find the quotients and any excluded values.
(x + 7)
x 2 + 9x + 14
_
÷ __
x2
x2 + x - 2
(x + 7) 2 __
(x + 7) 2
x2 + x - 2
x 2 + 9x + 14
_
÷ __
=_
⋅ 2
x + 9x + 14
x2
x2
x2 + x - 2
2
QUESTIONING STRATEGIES

© Houghton Mifflin Harcourt Publishing Company
Why must you exclude values of the variable
that make the numerator of the divisor 0? If
the numerator of a fraction is 0, then the fraction
equals 0. Since division by 0 is undefined, the
divisor cannot be equal to 0.
3x - 27
x ⋅_
_
x-9
x+1
3(x - 9)
x
= _
⋅ _______
(x - 9) x + 1
3x(x - 9)
= __
(x - 9)(x + 1)
(x + 3)(x - 3) _
x-8
____________
⋅
= ________
EXPLAIN 2
How is the procedure for dividing rational
expressions related to multiplying rational
expressions? Dividing by an expression is
equivalent to multiplying by its reciprocal. Once
division is converted to multiplication, you can carry
out the steps for multiplying rational expressions.
3.
Multiply by the reciprocal.
(x + 7)(x + 7) __
(x + 2)(x - 1)
= __
⋅
x2
(x + 7)(x + 2)
Factor the numerators and
denominators.
(x + 7)(x + 7)(x + 2)(x - 1)
= ___
x 2(x + 7)(x + 2)
Multiply the numerators and
multiply the denominators.
(x + 7)(x + 7)(x + 2)(x - 1)
= ___
2
x (x + 7)(x + 2)
Cancel the common factors in
the numerator and denominator.
(x + 7)(x - 1)
= __
x2
Determine what values of x make each expression undefined.
(x + 7) 2
_
:
x
The denominator is 0 when x = 0.
x + 9x + 14
__
: The denominator is 0 when x = -2 and x = 1.
x2 + x - 2
2
x2 + x - 2
__
: The denominator is 0 when x = -7 and x = -2.
x 2 + 9x + 14
Excluded values: x = 0, x = -7, x = 1, and x = -2
Module 9
441
Lesson 2
COLLABORATIVE LEARNING
A2_MNLESE385894_U4M09L2 441
Peer-to-Peer Activity
Have students work in pairs. Instruct each pair to create a problem involving the
division of two rational expressions by working backward from the factored form
of the numerators and denominators. Have them rewrite the problem, multiplying
the factors in each numerator and denominator. Then have them exchange
problems with another pair, and find the quotient. Have each pair compare their
answer to the answer determined by the students who created the problem.
441
Lesson 9.2
7/7/14 9:08 AM
B
9x 2 - 27x - 36
6x ÷ __
_
3x - 30
x 2 - 10x
x 2 - 10x
6x ⋅ __
6x ÷ __
9x 2 - 27x - 36 = _
_
3x - 30
3x - 30
x 2 - 10x
9x 2 - 27x - 36
(
)
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Prompt students to recognize that they can
Multiply by the reciprocal.
x x - 10
6x
= __
⋅ __ Factor the numerators and
denominators.
3 x - 10
9(x + 1) x - 4
(
)
6x (
)
(
)
2
x - 10
= ___
27 x - 10 (x + 1) x - 4
(
)
(
)
check their solutions to division problems by
multiplying the quotient by the divisor and
checking to see that the result is the dividend.
Multiply the numerators and
multiply the denominators.
2x 2
= __
9(x + 1)(x - 4)
Cancel the common factors in the
numerator and denominator.
Determine what values of x make each expression undefined.
6x
_
:
3x - 30
The denominator is 0 when x = 10 .
9x 2 - 27x - 36 : The denominator is 0 when x = 10 and x = 0 .
__
x 2 - 10x
x 2 - 10x
__
: The denominator is 0 when x = -1 and x = 4 .
9x 2 - 27x - 36
Excluded values:
x = 0, x = 10, x = -1, and x = 4
Your Turn
Find the quotients and any excluded values.
x + 11
2x + 6
_
÷_
4x
x 2 + 2x − 3
5.
20 ÷ __
5x 2 − 40x
_
2
x 2 − 7x
x − 15x + 56
∙ _________
= ______
4x
20
∙
= _______
2
x − 15x + 56
____________
(x − 1)(x + 3)
(x + 11) ____________
= _______
∙
4x
2(x + 3)
20
∙
= _______
(x − 8)(x − 7)
____________
x + 11
x 2 + 2x − 3
2x + 6
x − 7x
x(x − 7)
(x + 11)(x − 1)(x + 3)
= __________________
8x(x + 3 )
© Houghton Mifflin Harcourt Publishing Company
4.
2
5x 2 − 40x
5x(x − 8)
= ______________
2
20(x − 8)(x − 7)
5x (x − 7)(x − 8)
(x + 11)(x − 1)
= _____________
4
= __
2
Excluded values: x = 0, x = 1, and x = −3
Excluded values: x = 0, x = 7, and x = 8
8x
x
Module 9
Lesson 2
442
DIFFERENTIATE INSTRUCTION
A2_MNLESE385894_U4M09L2.indd 442
05/04/14 12:02 PM
Graphic Organizers
Have students copy and complete the graphic organizer shown below, writing a
worked-out example in each box.
Numerical Fractions
Rational Expressions
Adding
Subtracting
Multiplying
Dividing
Multiplying and Dividing Rational Expressions
442
Explain 3
EXPLAIN 3
Activity: Investigating Closure
A set of numbers is said to be closed, or to have closure, under a given operation if the result of the operation on any
two numbers in the set is also in the set.
Activity: Investigating Closure
A
AVOID COMMON ERRORS
Students may think that a single example is sufficient
to prove that a set is closed. While a single
counterexample is enough to prove that a set is not
closed, the general result must be proven to show
closure. For example, the quotient of the integer
division 8 ÷ 2 = 4 is an integer, but the integers are
not closed under division.
B
Recall whether the set of whole numbers, the set of integers, and the set of rational numbers
are closed under each of the four basic operations.
Addition
Subtraction
Multiplication
Division
Whole Numbers
Closed
Not Closed
Closed
Not Closed
Integers
Closed
Closed
Closed
Not Closed
Rational Numbers
Closed
Closed
Closed
Closed
p(x)
r(x)
Look at the set of rational expressions. Use the rational expressions ___
and ___
where p(x),
q(x)
s(x)
q(x), r(x) and s(x) are nonzero. Add the rational expressions.
p(x)
r(x)
p(x)s(x) + q(x)r(x)
_
+ _ =
q(x)s(x)
q(x)
s(x)
______________
C
Is the set of rational expressions closed under addition? Explain.
p(x)s(x) + q(x)r(x)
Yes; since q(x) and s(x) are nonzero, q(x)s(x) is nonzero. So, ______________ is again a
q(x)s(x)
QUESTIONING STRATEGIES
rational expression.
How do you determine whether a set of
polynomials or rational expressions is closed
under a given operation? Define the members of the
set. Then investigate the set to determine whether
the given operation always results in a member of
the set.
counter example to show that a set is not closed than
to explain why a set is closed. Encourage students to
use variables such as a and b to represent elements of
the set, and try to determine the general result of the
operation on a and b.
Subtract the rational expressions.
p(x)
r(x)
p(x)s(x) - q(x)r(x)
_
- _ =
q(x)
s(x)
q(x)s(x)
______________
E
Is the set of rational expressions closed under subtraction? Explain.
p(x)s(x) - q(x)r(x)
Yes; since q(x) and s(x) are nonzero, q(x)s(x) is nonzero. So, ______________ is again a
q(x)s(x)
rational expression.
© Houghton Mifflin Harcourt Publishing Company
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 For most students, it will be easier to give a
D
F
Multiply the rational expressions.
p(x) _
r(x)
p(x)r(x)
_
∙
=
q(x)s(x)
q(x)
s(x)
G
Is the set of rational expressions closed under multiplication? Explain.
p(x)r(x)
Yes; since q(x) and s(x) are nonzero, q(x)s(x) is nonzero. So,
is again a rational
______
______
q(x)s(x)
expression.
H
Divide the rational expressions.
p(x)s(x)
p(x)
r(x)
_
÷ _=
q(x)r(x)
q(x)
s(x)
I
Is the set of rational expressions closed under division? Explain.
p(x)s(x)
Yes; since q(x) and r(x) are nonzero, q(x)r(x) is nonzero. So, ______ is again a rational
______
q(x)r(x)
expression.
Module 9
A2_MNLESE385894_U4M09L2 443
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Lesson 9.2
443
Lesson 2
7/7/14 9:10 AM
Reflect
6.
EXPLAIN 4
Are rational expressions most like whole numbers, integers, or rational numbers? Explain.
Rational expressions are like rational numbers because both the set of rational
Multiplying and Dividing with
Rational Models
expressions and the set of rational numbers are closed under all four basic operations.
Explain 4
Multiplying and Dividing with Rational Models
Models involving rational expressions can be solved using the same steps to multiply or divide rational expressions.
Example 3

QUESTIONING STRATEGIES
Solve the problems using rational expressions.
How do you determine the excluded values in
a real-world problem that involves dividing
two rational expressions? Find the values that make
each denominator 0 and that make the numerator
of the divisor 0. Also, determine numbers that are
not reasonable values for the independent variable
in the situation.
Leonard drives 40 miles to work every day. One-fifth of his
drive is on city roads, where he averages 30 miles per hour.
The other part of his drive is on a highway, where
d cr h + d hr c
he averages 55 miles per hour. The expression ________
r cr h
represents the total time spent driving, in hours. In the
expression, d c represents the distance traveled on city
roads, d h represents the distance traveled on the highway,
r c is the average speed on city roads, and r h is the average
speed on the highway. Use the expression to find the
average speed of Leonard’s drive.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Discuss with students how the rational
The total distance traveled is 40 miles. Find an expression
for the average speed, r, of Leonard’s drive.
r cr h
= 40 ∙ _
d cr h + d hr c
40r cr h
=_
d cr h + d hr c
Find the values of d c and d h .
1 (40) = 8 miles
dc = _
5
d h = 40 - 8 = 32 miles
Solve for r by substituting in the given values from the
problem.
dr cr h
r= _
d cr h + d hr c
40 ∙ 55 ∙ 30
= __
8 ∙ 55 + 32 ∙ 30
≈ 47 miles per hour
The average speed of Leonard’s drive is about 47 miles per hour.

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ocean/
Corbis
r = Total distance traveled ÷ Total time
d cr h + d hr c
= 40 ÷ _
r cr h
expressions used in the example model the situation.
Discuss what each numerator and denominator
represents, and why a quotient of these quantities is
an appropriate model.
The fuel efficiency of Tanika’s car at highway speeds is 35 miles per gallon. The
- 216
________
expression 48E
represents the total gas consumed, in gallons, when Tanika drives
E( E - 6 )
36 miles on a highway and 12 miles in a town to get to her relative’s house. In the
expression, E represents the fuel efficiency, in miles per gallon, of Tanika’s car at highway
speeds. Use the expression to find the average rate of gas consumed on her trip.
Module 9
A2_MNLESE385894_U4M09L2.indd 444
444
Lesson 2
25/03/14 12:43 AM
Multiplying and Dividing Rational Expressions
444
The total distance traveled is 48 miles. Find an
expression for the average rate of gas consumed, g,
on Tanika’s trip.
ELABORATE
g = Total gas consumed ÷ Total distance traveled
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Call upon students to describe each step
48E - 216 ÷
=_
E(E - 6)
48
(
)
35
48
- 216
g = ___
48 35
35 - 6
(
)(
)
1464
=_
48,720
48E - 216
= __
48 E (E - 6)
involved in the solution to a problem involving
division of two rational expressions. Make sure they
use accurate mathematical language in describing not
only the division process, but also how to identify
excluded values of the variable.
Solve for g by substituting in the value of E.
≈ 0.03
The value of E is 35 .
The average rate of gas consumed on Tanika’s
trip is about 0.03 gallon per mile.
Your Turn
7.
SUMMARIZE THE LESSON
How do you divide two rational
expressions? Multiply the first rational
expression by the reciprocal of the second. Factor
each numerator and denominator, and then
multiply numerators and multiply denominators.
Divide out common factors of the numerators and
denominators.
The distance traveled by a car undergoing constant acceleration, a, for a time, t, is given by d = v 0t +
1 2
_
at , where v 0 is the initial velocity of the car. Two cars are side by side with the same initial velocity. One
2
car accelerates and the other car does not. Write an expression for the ratio of the distance traveled by the
accelerating car to the distance traveled by the nonaccelerating car as a function of time.
Let A be the acceleration of the accelerating car.
Accelerating car:
1 2
At
d = v 0t + __
2
Nonaccelerating car:
1( )
d = v t + __
0 t
0
2
0
= v 0t
2
v 0t
Distance of nonaccelerating car
2
__1
v t + 2 At
Distance of accelerating car
_________________________
= ________
__1 At
2
0
= ___
+ ____
v 0t
v 0t
vt
2
At
= 1 + ___
2v
0
At
The ratio as a function of time is 1 + ___
.
2v
© Houghton Mifflin Harcourt Publishing Company
0
Elaborate
8.
Explain how finding excluded values when dividing one rational expression by another is different from
multiplying two rational expressions.
When finding excluded values of a product of two rational expressions, find the values
of x for which the denominator of either expression is 0. When finding excluded values
when dividing one rational expression by another, find the values of x for which the
denominator of either expression or the numerator of the second expression is 0.
9.
Essential Question Check-In How is dividing rational expressions related to multiplying rational
expressions?
When dividing rational expressions, find the reciprocal of the divisor and change the
division problem to a multiplication problem. Then follow the steps for multiplying
rational expressions.
Module 9
445
Lesson 2
LANGUAGE SUPPORT
A2_MNLESE385894_U4M09L2.indd 445
Communicate Math
Have students work in pairs. Provide each pair of students with some rational
expressions to multiply or divide, written on sticky notes or index cards. Have the
first student explain the steps to multiply rational expressions while the second
student writes notes. Students switch roles and repeat the process for a division
problem, highlighting the additional step of using the reciprocal.
445
Lesson 9.2
6/10/15 12:42 AM
EVALUATE
Evaluate: Homework and Practice
1.
• Online Homework
• Hints and Help
• Extra Practice
Explain how to multiply the rational expressions.
2
- 3x
+4
x - 3 ⋅ x_
_
_
2
x 2 - 2x
Multiply x - 3 by x 2 - 3x + 4 to get the numerator of the product. Multiply 2 by x 2 - 2x to
get the denominator of the product. Then, simplify by cancelling common factors in the
numerator and the denominator.
Find the products and any excluded values.
x
x-2
⋅_
2. _
3x - 6 x + 9
ASSIGNMENT GUIDE
3.
x-2
x
⋅_
=_
3(x - 2) x + 9
x(x - 2)
= __
3(x - 2)(x + 9)
x
=_
3(x + 9)
x - 2x - 15 ⋅ __
3
__
10x + 30
x 2 - 3x - 10
2
(x - 5)(x + 3)
3
= __ ⋅ __
10(x + 3)
(x + 2)(x - 5)
3(x - 5)(x + 3)
= ___
10(x + 3)(x + 2)(x - 5)
Concepts and Skills
Practice
5x(x + 5)
4x
=_⋅_
2
x+5
Explore
Relating Multiplication Concepts
Exercise 1
20x 2(x + 5)
= __
2(x + 5)
Example 1
Multiplying Rational Expressions
Exercises 2–7
= 10x 2
Example 2
Dividing Rational Expressions
Exercises 8–13
Example 3
Activity: Investigating Closure
Exercises 14–17
Example 4
Multiplying and Dividing with
Rational Models
Exercises 18–20
Excluded value: x = -5
Excluded values: x = 2 and x = -9
4.
5x 2 + 25x _
_
⋅ 4x
x+5
2
5.
x2 - 1
x2
__
⋅_
x 2 + 5x + 4 x 2 - x
(x - 1)(x + 1)
x2
= __ ⋅ _
(x + 4)(x + 1) x(x - 1)
(x - 1)(x + 1)x 2
= ___
x(x + 4)(x + 1)(x - 1)
x
=_
x+ 4
and x = 5
x = 0, and x = 1
Excluded values: x = -3, x = -2,
Module 9
Excluded values: x = -4, x = -1,
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Practices
1
1 Recall of Information
MP.6 Precision
2–13
1 Recall of Information
MP.2 Reasoning
14–17
2 Skills/Concepts
MP.6 Precision
18–20
2 Skills/Concepts
MP.4 Modeling
21–22
3 Strategic Thinking
MP.2 Reasoning
2 Skills/Concepts
MP.4 Modeling
23
When multiplying rational expressions, students may
divide out by common factors and then, erroneously,
cross-multiply instead of multiplying straight across.
Remind them that cross-multiplying is used to solve
equations, and that when multiplying two rational
expressions, they must multiply straight across.
Lesson 2
446
A2_MNLESE385894_U4M09L2 446
AVOID COMMON ERRORS
© Houghton Mifflin Harcourt Publishing Company
3
=_
10(x + 2)
7/7/14 9:13 AM
Multiplying and Dividing Rational Expressions
446
6.
AVOID COMMON ERRORS
2
x 2 + 14x + 33 _
8x - 56
__
⋅ x - 3x ⋅ __
4x
x + 3 x 2 + 4x - 77
7.
3
9x 2 (x + 6)(x - 6)
= _ ⋅ __ ⋅ _
x-6
4x(x + 6)
3(x - 2)
(x + 11)(x + 3) x(x - 3)
8(x - 7)
= __ ⋅ _ ⋅ __
4x
x+3
(x + 11)(x - 7)
When identifying excluded values for quotients of
rational expressions, students may consider values
that cause the denominators to be zero, but they may
forget to consider values that cause the divisor itself
x 2 - 36
will have
to be 0. For example, the divisor ______
x 2 - 4x
a value of 0 when x = 6 or x = -6, so these values
must also be excluded values.
9x 2 ⋅ _
x 2 - 36 ⋅ _
3
_
x - 6 3x - 6 4x 2 + 24x
27x 2 (x + 6)(x - 6)
= ___
12x(x - 6)(x - 2)(x + 6)
8x(x + 11)(x + 3)(x - 3)(x - 7)
= ___
4x(x + 3)(x + 11)(x - 7)
= 2(x - 3)
9x
=_
4(x - 2)
Excluded values: x = 0, x = -3, x = -11, and
Excluded values: x = 6, x = 2, x = 0
and x = -6
x=7
Find the quotients and any excluded values.
20x + 40
5x 2 + 10x
÷_
8. _
x 2 + 2x + 1
x2 - 1
9.
x 2 - 9x + 18
x 2 - 36
__
÷_
x 2 + 9x + 18
x2 - 9
=
5x + 10x _
x -1
__
⋅
=
x - 9x + 18 _
x -9
__
⋅
=
5x(x + 2)
(x + 1)(x - 1)
__
⋅ __
=
(x + 3)(x - 3)
(x - 6)(x - 3) __
__
⋅
5x(x + 2)(x + 1)(x - 1)
___
=
x(x - 1)
_
=
=
=
2
x 2 + 2x + 1
2
20x + 40
20(x + 2)
(x + 1)(x + 1)
20 (x + 1)(x + 1)(x + 2)
4(x + 1)
2
2
x 2 + 9x + 18
x 2 - 36
(x + 6)(x + 3) (x + 6)(x - 6)
(x - 6)(x - 3)(x + 3)(x - 3)
___
(x + 6)(x + 3)(x + 6)(x - 6)
(x - 3)
_
2
(x + 6)2
Excluded values: x = ±6, x = ±3
Excluded values: x = 1, x = -1, and
© Houghton Mifflin Harcourt Publishing Company
x = -2
-x 2 + x + 20
x+4
10. __
÷_
2x - 14
5x 2 - 25x
x+3
x 2 - 25
11. __
÷_
x-5
x 2 + 8x + 15
-x + x + 20 _
2x - 14
__
⋅
2
=
5x 2 - 25x
x+4
-(x + 4)(x - 5) 2(x - 7)
= __ ⋅ _
5x(x - 5)
x+4
-2(x + 4)(x - 5)(x - 7)
= ___
5x(x - 5)(x + 4)
_
Lesson 9.2
x+3
x-5
__
⋅ __
x 2 + 8x + 15
x 2 - 25
(x + 5)(x + 3) (x + 5)(x - 5)
(x + 3)(x - 5)
___
(x + 5)(x + 3)(x + 5)(x - 5)
1
_
(x + 5) 2
Excluded values: x = -5, x = -3, and
x = -4
447
=
=
Excluded values: x = 0, x = 5, x = 7 and
A2_MNLESE385894_U4M09L2 447
x+3
x-5
__
⋅_
=
-2(x - 7)
=
5x
Module 9
=
x=5
447
Lesson 2
7/7/14 9:15 AM
x 2 - 10x + 9
x 2 - 7x - 18
12. __ ÷ __
3x
x 2 + 2x
=
x + 2x
x - 10x + 9 __
__
⋅
=
x(x + 2)
(x - 1)(x - 9) __
__
⋅
2
8x + 32
x 2 - 6x
13. __
÷ __
x 2 + 8x + 16
x 2 - 2x - 24
2
3x
3x
x - 7x - 18
2
(x + 2)(x - 9)
x(x - 1)(x - 9)(x + 2)
= ___
3x(x + 2)(x - 9)
=
=
8x + 32
x - 2x - 24
__
⋅ __
=
8(x + 4)
(x + 4)(x - 6)
__
⋅ __
=
x-1
_
SMALL GROUP ACTIVITY
Have students work in small groups to make a poster
showing how to divide two rational expressions. Give
each group a different problem, each consisting of
polynomials that require several different factoring
strategies. Then have each group present its poster to
the rest of the class, explaining each step.
2
x 2 + 8x + 16
x 2 - 6x
(x + 4)(x + 4)
x(x - 6)
8(x + 4)(x + 4)(x - 6)
___
x(x + 4)(x + 4)(x - 6)
_
8
= x
3
Excluded values: x = 0, x = -2, and
Excluded values: x = 0, x = -4, and x = 6
x=9
1
1
Let p(x) = ____
and q(x) = ____
. Perform the operation, and show
x+1
x-1
that it results in another rational expression.
14. p(x) + q(x)
2x
__________
; the numerator and denominator are polynomials, so it is
(x + 1)(x - 1)
a rational expression.
15. p(x) - q(x)
-2
__________
; the numerator and denominator are polynomials, so it is
(x + 1)(x - 1)
a rational expression.
© Houghton Mifflin Harcourt Publishing Company
16. p(x) ⋅ q(x)
1
__________
; the numerator and denominator are polynomials, so it is
(x + 1)(x - 1)
a rational expression.
17. p(x) ÷ q(x)
x-1
____
; the numerator and denominator are polynomials, so it is
x+1
a rational expression.
Module 9
A2_MNLESE385894_U4M09L2 448
448
Lesson 2
6/15/15 11:31 AM
Multiplying and Dividing Rational Expressions
448
18. The distance a race car travels is given by the equation
d = v 0t + __12 at 2, where v 0 is the initial speed of the race car,
a is the acceleration, and t is the time travelled. Near the
beginning of a race, the driver accelerates for 9 seconds
at a rate of 4 m/s 2. The driver’s initial speed was 75 m/s.
Find the driver’s average speed during the acceleration.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Students can use a graphing calculator to
The average speed is equal to the
compare the graph of the function defined by the
original product or quotient with the graph of the
function defined by the final simplified expression. If
the expressions are equivalent, the graphs should be
identical.
distance traveled divided by the time.
d
r=
t
1
t v 0 + at
2
=
t
1
= v 0 + at
2
_
( _ )
__
_
Substitute the known values
into the equation to find r.
1
r = v 0 + at
2
1
= 75 + (4)(9)
2
_
_
= 93
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David
Madison/Corbis
The average speed during the
acceleration is 93 meters per
second.
19. Julianna is designing a circular track that will consist of three concentric rings.
The radius of the middle ring is 6 meters greater than that of the inner ring and
6 meters less than that of the outer ring. Find an expression for the ratio of the length
of the outer ring to the length of the middle ring and another for the ratio of the
length of the outer ring to length of the inner ring. If the radius of the inner ring is
set at 90 meters, how many times longer is the outer ring than the middle ring and
the inner ring?
Length of inner ring: 2πr
Length of middle track: 2π(r + 6)
Length of outer ring: 2π(r + 12)
2π(r + 12)
Length of outer ring
__
= __
r + 12
=_
r+6
2π(r + 12)
Length of outer ring
__
= __
=
A2_MNLESE385894_U4M09L2 449
Lesson 9.2
2πr
Length of inner ring
Module 9
449
2π(r + 6)
Length of middle ring
r + 12
_
r
Substitute 90 for r.
90 + 12 _
102
_
=
= 1.0625
90 + 6
96
90
90
90 + 12 _
102
_
=
≈ 1.13
The outer ring is 1.0625 times longer
than the middle ring and about 1.13
times longer than the inner ring.
449
Lesson 2
7/7/14 9:20 AM
MODELING
20. Geometry Find a rational expression for the ratio of the surface area of a cylinder
to the volume of a cylinder. Then find the ratio when the radius is 3 inches and the
height is 10 inches.
Surface Area = 2πr 2 + 2πrh
Substitute 3 for r and 10 for h.
2
(3 + 10)
_______
Volume = πr 2h
(3)(10)
2πr + 2πrh
Surface Area
_________
= ________
2
26
13
= __
= __
15
30
The ratio of the cylinder’s surface area
πr 2h
Volume
When working with rational expressions that
represent real-world situations, students should
recognize that not only must they consider excluded
values that are based on the algebraic nature of the
rational expressions, but they also need to consider
values that must be excluded due to the limitations
on the domain in the given situation.
2πr(r + h)
= _______
2
to its volume is 13:15.
πr h
2(r + h)
= ______
rh
H.O.T. Focus on Higher Order Thinking
21. Explain the Error Maria finds an
expression equivalent to
6x 2 - 150 .
x 2 - 4x - 45 ÷ _
__
x 2 - 5x
3x - 15
Her work is shown. Find
and correct Maria’s mistake.
CONNECT VOCABULARY
(x - 9)(x + 5)
6(x + 5)(x - 5)
x 2 - 4x - 45
6x 2 - 150
__
= __ ÷ __
÷_
3x - 15
x 2 - 5x
(
)
3 x-5
x(x - 5)
Have students complete a vocabulary chart using
rational numbers and rational expressions, with
examples of both fractions and rational expressions.
Include the terms used in this lesson: numerator,
denominator, factor, reciprocal.
6(x - 9)(x + 5)(x + 5)(x - 5)
= ___
3x(x - 5)(x - 5)
2
2(x - 9)(x + 5)
= __
(
)
x x-5
Maria did not multiply by the reciprocal.
2
2
2
x________
- 4x - 45
x 2 - 150
- 4x - 45 _______
÷ 6_______
= x________
⋅ x 2 - 5x
3x - 15
3x - 15
x2 - 5x
6x - 150
(x - 9)(x + 5)
x(x - 5)
= _________ ⋅ __________
3(x - 5)
6(x + 5)(x - 5)
=
x(x - 9)(x + 5)(x - 5)
________________
18x(x - 5)(x + 5)(x - 5)
x(x - 9)
18(x - 5)
© Houghton Mifflin Harcourt Publishing Company
= _______
22. Critical Thinking Multiply the expressions. What do you notice about the resulting expression?
3x + 18
3
x
x - 4x
+ _ )( __ - _)
(_
x-4
8
8x - 32 x + 2x - 24
3
2
2
(
x(x + 2)(x- 2)
3
+ __________
= ____
x-4
8(x + 2)(x - 2)
(
)(
)
3
3
x ____
x
+_
-_
= ____
x-4
8 x-4
8
(
3
= ____
x-4
) - (_8x)
2
)(
)
3(x + 6)
x
___________
-_
8
(x - 4)(x + 6)
2
3
x _
3
= _____
∙ ____
-_
∙x
x-4 x-4
8 8
x
9
_
= ______
2 64
2
(x - 4)
The expression is the difference of two squares.
Module 9
A2_MNLESE385894_U4M09L2 450
450
Lesson 2
16/10/14 1:59 PM
Multiplying and Dividing Rational Expressions
450
23. Multi-Step Jordan is making a garden with an area
of x 2 + 13x + 30 square feet and a length of x + 3 feet.
JOURNAL
Have students compare and contrast the method they
have learned for multiplying rational expressions
with the method they have learned for adding
rational expressions.
a. Find an expression for the width of Jordan’s garden.
+ 13x + 30 ____
(x 2 + 13x + 30) ÷ (x + 3) = x_________
⋅ x +1 3
1
2
(x + 10)(x + 3) ____
1
⋅ x+
= __________
1
3
(x + 10)(x + 3)
= __________
x+3
= x + 10
Jordan’s garden is x + 10 feet wide.
b. If Karl makes a garden with an area of 3x 2 + 48x + 180 square feet and a length of x + 6, how
many times larger is the width of Jon’s garden than Jordan’s?
x + 48x + 180 ____
(3x 2 + 48x + 180) ÷ (x + 6) = 3___________
⋅ x +1 6
1
2
3(x + 10)(x + 6) ____
1
⋅ x+
= ___________
1
6
3(x + 10)(x + 6)
= ___________
x+6
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Shannon
Fagan/Corbis
= 3(x + 10)
Karl’s garden is 3 times wider than Jordan’s garden.
c. If x is equal to 4, what are the dimensions of both Jordan’s and Karl’s gardens?
Jordan’s garden:
Length: 4 + 3 = 7 feet
Width: 4 + 10 = 14 feet
Karl's garden:
Length: 4 + 6 = 10 feet
Width: 3(4 + 10) = 42 feet
Module 9
A2_MNLESE385894_U4M09L2 451
451
Lesson 9.2
451
Lesson 2
7/7/14 9:31 AM
Lesson Performance Task
AVOID COMMON ERRORS
Who has the advantage, taller or shorter runners? Almost all of the energy generated by a
long-distance runner is released in the form of heat. For a runner with height H and
speed V, the rate h g of heat generated and the rate h r of heat released can be modeled by
h g = k 1H 3V 2 and h r = k 2H 2, k 1 and k 2 being constants. So, how does a runner’s height affect
the amount of heat she releases as she increases her speed?
Students may think that the amount of heat released
by the runner is independent of speed because
h r = k 2 H2, which is independent of V. Ask students
where the heat comes from before it is
released. generated by runner Then ask what the
expression is for the heat generated, and ask whether
it depends on V. h g = k 1 H 3V 2 ; yes Then have
students write the equation for the situation that
occurs when the amount of heat released is equal to
the amount of heat generated.
First, set up the ratio for the amount of heat generated by the runner to
the amount of heat released by dividing the value h g by h r.
h
kHV
__
= _____
g
hr
1
3
2
k 2H 2
Next, simplify the ratio.
h
k HV
__
= ____
g
1
2
k2
hg
is equal to 1, the amount of heat released is the same as
When __
hr
hr
the amount of heat generated. You can use this condition as a way to
determine the relationship of height to speed. Setting the ratio equal to 1,
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Have students draw a graph for the relation
isolate speed on one side of the equation.
k
___
= V2
2
k 1H
Since k 1 and k 2 are constants, you see that as a runner’s height increases,
the speed required to maintain the balance of heat generated to heat
they obtained between height and speed. Ask them
what information they need to draw the exact graph
for the relation. the constants k 1 and k 2 Then, have
students discuss whether this graph helps them
determine an ideal height for a runner.
released gets smaller. Therefore, a shorter runner can run at a higher
speed and not lose as much heat as a taller runner does.
© Houghton Mifflin Harcourt Publishing Company
Module 9
452
Lesson 2
EXTENSION ACTIVITY
A2_MNLESE385894_U4M09L2 452
Ask students to rework the problem, this time with the heat generated modeled by
h g = k 1 H 3V . Ask them to describe the relation between speed and height, and to
tell how that relation differs from the answer they calculated in the Performance
Task. Ask them whether this model gives shorter runners a greater or lesser
advantage, compared to the model in the Performance Task. Shorter runners still
have an advantage over taller runners, but it is not as great.
7/7/14 9:33 AM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Multiplying and Dividing Rational Expressions
452