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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Lesson 24: Multiplying and Dividing Rational Expressions
Student Outcomes

Students multiply and divide rational expressions and simplify using equivalent expressions.
Lesson Notes
This lesson quickly reviews the process of multiplying and dividing rational numbers using techniques students already
know and translates that process to multiplying and dividing rational expressions (MP.7). This enables students to
develop techniques to solve rational equations in Lesson 26 (A-APR.D.6). This lesson also begins developing facility with
simplifying complex rational expressions, which is important for later work in trigonometry. Teachers may consider
treating the multiplication and division portions of this lesson as two separate lessons.
Classwork
Opening Exercise (5 minutes)
Distribute notecard-sized slips of paper to students, and ask them to shade the paper to
represent the result of
2 4
⋅ . Circulate around the classroom to assess student proficiency.
3 5
If many students are still struggling to remember the area model after the scaffolding,
present the problem to them as shown. Otherwise, allow them time to do the
multiplication on their own or with their neighbor and then progress to the question of the
general rule.

Scaffolding:
If students do not remember
the area model for
multiplication of fractions,
have them discuss it with their
neighbor. If necessary, use an
1 1
⋅
4
example like
5
can scale this to the problem
presented.
First, we represent by dividing our region into five vertical strips of equal area
and shading 4 of the 5 parts.
2 2
to see if they
If students are comfortable
with multiplying rational
numbers, omit the area model,
and ask them to determine the
following products.

2
Now we need to find of the shaded area. So we divide the area horizontally
3



2 3
∙
=
∙
=
∙
=
3 8
1 5
4 6
4 8
7 9
1
4
5
24
32
63
into three parts of equal area and then shade two of those parts.
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
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M1
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
ALGEBRA II

Thus,
2 4
∙
3 5
is represented by the region that is shaded twice. Since 8 out of 15 subrectangles are shaded
twice, we have
2 4
∙
3 5
=
8
15
. With this in mind, can we create a general rule about multiplying rational
numbers?
Allow students to come up with this rule based on the example and prior experience. Have them discuss their thoughts
with their neighbor and write the rule.
If 𝑎, 𝑏, 𝑐, and 𝑑 are integers with 𝑐 ≠ 0 and 𝑑 ≠ 0, then
𝑎 𝑏 𝑎𝑏
∙ =
.
𝑐 𝑑 𝑐𝑑
The rule summarized above is also valid for real numbers.
Discussion (2 minutes)

To multiply rational expressions, we follow the same procedure we use when multiplying rational numbers:
we multiply together the numerators and multiply together the denominators. We finish by reducing the
product to lowest terms.
If 𝒂, 𝒃, 𝒄, and 𝒅 are rational expressions with 𝒃 ≠ 𝟎, 𝒅 ≠ 𝟎, then
𝒂 𝒄 𝒂𝒄
∙ =
.
𝒃 𝒅 𝒃𝒅
Lead students through Examples 1 and 2, and ask for their input at each step.
Scaffolding:
Example 1 (4 minutes)
Give students time to work on this problem and discuss their answers with a
neighbor before proceeding to class discussion.
Example 1
Make a conjecture about the product
𝒙𝟑
𝟒𝒚
∙
𝒚𝟐
𝒙
We begin by multiplying the numerators and denominators.
𝑥 3 𝑦2 𝑥 3𝑦2
∙
=
4𝑦 𝑥
4𝑦𝑥
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
𝑥3
2 4
8
⋅ =
3 5 15
. What will it be? Explain your
conjecture, and give evidence that it is correct.

 To assist students in making the
connection between rational numbers
and rational expressions, show a sideby-side comparison of a numerical
example from a previous lesson like the
one shown.
4𝑦
⋅
𝑦2
𝑥
=?
 If students are struggling with this
example, include some others, such as
𝑥2
3
⋅
6
𝑥
OR
𝑦
𝑥2
⋅
𝑦4
𝑥
.
269
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II

Identify the greatest common factor (GCF) of the numerator and denominator. The GCF of 𝑥 3 𝑦 2 and 4𝑥𝑦 is
𝑥𝑦.
𝑥 3 𝑦 2 (𝑥𝑦)𝑥 2 𝑦
∙
=
4𝑦 𝑥
4(𝑥𝑦)

Finally, we divide the common factor 𝑥𝑦 from the numerator and denominator to find the reduced form of the
product:
𝑥 3 𝑦2 𝑥 2𝑦
∙
=
.
4𝑦 𝑥
4
Note that the phrases “cancel 𝑥𝑦” or “cancel the common factor” are intentionally avoided in this lesson. The goal is to
highlight that it is division that allows these expressions to be simplified. Ambiguous words like “cancel” can lead
students to simplify
sin(𝑥)
𝑥
to sin—they “canceled” the 𝑥!
It is important to understand why the numerator and denominator may be divided by 𝑥. The rule
rational expressions as well. Performing a simplification such as
𝑥
𝑥3 𝑦
=
𝑥∙1
𝑥∙𝑥 2 𝑦
=
𝑥
𝑥
⋅
1
𝑥2 𝑦
=1⋅
1
𝑥2 𝑦
=
1
𝑥2 𝑦
𝑥
𝑥3 𝑦
=
1
𝑥2 𝑦
𝑛𝑎
𝑛𝑏
=
𝑎
𝑏
works for
requires doing the following steps:
.
Example 2 (3 minutes)
Before walking students through the steps of this example, ask them to try to find the product using the ideas of the
previous example.
Example 2
Find the following product: (
𝟑𝒙−𝟔
𝟓𝒙+𝟏𝟓
)∙(
).
𝟐𝒙+𝟔
𝟒𝒙+𝟖
First, factor the numerator and denominator of each rational expression.

Identify any common factors in the numerator and denominator.
3𝑥 − 6
5𝑥 + 15
3(𝑥 − 2)
5(𝑥 + 3)
(
)∙(
)=(
)∙(
)
2𝑥 + 6
4𝑥 + 8
2(𝑥 + 3)
4(𝑥 + 2)
MP.7
=
15(𝑥 − 2)(𝑥 + 3)
8(𝑥 + 3)(𝑥 + 2)
The GCF of the numerator and denominator is 𝑥 + 3.
Then, divide the common factor (𝑥 + 3) from the numerator and denominator, and obtain the reduced form of the
product.
3𝑥 − 6
5𝑥 + 15
15(𝑥 − 2)
(
)∙(
)=
2𝑥 + 6
4𝑥 + 8
8(𝑥 + 2)
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
270
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
M1
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
ALGEBRA II
Exercises 1–3 (5 minutes)
Students can work in pairs on the following three exercises. Circulate around the class to informally assess their
understanding. For Exercise 1, listen for key points such as “factoring the numerator and denominator can help” and
“multiplying rational expressions is similar to multiplying rational numbers.”
Exercises 1–3
1.
Summarize what you have learned so far with your neighbor.
Answers will vary.
2.
Find the following product and reduce to lowest terms: (
𝟐𝒙+𝟔
𝒙𝟐 −𝟒
)∙(
).
𝟐𝒙
𝒙𝟐 +𝒙−𝟔
(𝒙 − 𝟐)(𝒙 + 𝟐)
𝟐𝒙 + 𝟔
𝒙𝟐 − 𝟒
𝟐(𝒙 + 𝟑)
( 𝟐
)∙(
)=(
)∙(
)
(𝒙 + 𝟑)(𝒙 − 𝟐)
𝒙 +𝒙−𝟔
𝟐𝒙
𝟐𝒙
=
𝟐(𝒙 + 𝟑)(𝒙 − 𝟐)(𝒙 + 𝟐)
𝟐𝒙(𝒙 + 𝟑)(𝒙 − 𝟐)
The factors 𝟐, 𝒙 + 𝟑, and 𝒙 − 𝟐 can be divided from the numerator and the denominator in order to reduce the
rational expression to lowest terms.
𝟐𝒙 + 𝟔
𝒙𝟐 − 𝟒
𝒙+𝟐
( 𝟐
)∙(
)=
𝒙 +𝒙−𝟔
𝟐𝒙
𝒙
3.
Find the following product and reduce to lowest terms: (
Scaffolding:
Students may need to be reminded
how to interpret a negative
exponent. If so, ask them to
calculate these values.
𝟒𝒏−𝟏𝟐 −𝟐
𝒏𝟐 −𝟐𝒏−𝟑
) ∙( 𝟐
).
𝟑𝒎+𝟔
𝒎 +𝟒𝒎+𝟒
𝟒𝒏 − 𝟏𝟐 −𝟐
𝒏𝟐 − 𝟐𝒏 − 𝟑
𝟑𝒎 + 𝟔 𝟐
𝒏𝟐 − 𝟐𝒏 − 𝟑
(
) ∙( 𝟐
) ∙( 𝟐
)=(
)
𝟑𝒎 + 𝟔
𝒎 + 𝟒𝒎 + 𝟒
𝟒𝒏 − 𝟏𝟐
𝒎 + 𝟒𝒎 + 𝟒
 3−2 =
𝟑𝟐 (𝒎 + 𝟐)𝟐 (𝒏 − 𝟑)(𝒏 + 𝟏)
=
𝟒𝟐 (𝒏 − 𝟑)𝟐 (𝒎 + 𝟐)𝟐
𝟗(𝒏 + 𝟏)
=
𝟏𝟔(𝒏 − 𝟑)
1
3
2
=
1
9
3

2 −3
5 3 5
125
( ) =( ) = 3=
5
2
8
2

𝑥
( 2)
𝑦
−5
5
5
(𝑦2 )
𝑦2
𝑦10
=( ) = 5 = 5
𝑥
𝑥
𝑥
Discussion (5 minutes)
Recall that division of numbers is equivalent to multiplication of the numerator by the
reciprocal of the denominator. That is, for any two numbers 𝑎 and 𝑏, where 𝑏 ≠ 0, we
have
𝑎
1
=𝑎∙ ,
𝑏
𝑏
where the number
1
𝑏
is the multiplicative inverse of 𝑏. But, what if 𝑏 is itself a fraction?
How do we evaluate a quotient such as

How do we evaluate
3
5
3
5
4
÷ ?
7
4
÷ ?
7
Have students work in pairs to answer this and then discuss.
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
Scaffolding:
Students may be better able to
generalize the procedure for
dividing rational numbers by
repeatedly dividing several
examples, such as
2
3
÷
7
10
, and
1
5
2
1
2
3
÷ ,
4
÷ . After
9
dividing several of these
examples, ask students to
generalize the process (MP.8).
271
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M1
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
ALGEBRA II
By our rule above,
3
5
÷
4
7
3
=
5
1
1
1
⁄7
⁄7
⁄7
∙ 4 . But, what is the value of 4 ? Let 𝑥 represent 4 , which is the multiplicative
4
inverse of . Then we have
7
4
=1
7
4𝑥 = 7
7
𝑥= .
4
𝑥∙
1
Since we have shown that 4
⁄7
7
3
4
5
= , we can continue our calculation of
÷
4
7
as follows:
3 4 3 1
÷ = ∙
5 7 5 4⁄
7
3 7
= ∙
5 4
21
=
.
20
This same process applies to dividing rational expressions, although we might need to perform the additional step of
reducing the resulting rational expression to lowest terms. Ask students to generate the rule for division of rational
numbers.
If 𝑎, 𝑏, 𝑐, and 𝑑 are integers with 𝑏 ≠ 0, 𝑐 ≠ 0, and 𝑑 ≠ 0, then
𝑎 𝑏 𝑎 𝑑
÷ = ∙ .
𝑐 𝑑 𝑐 𝑏
The result summarized in the box above is also valid for real numbers.
Scaffolding:
Now that we know how to divide rational numbers, how do we extend this to
divide rational expressions?
A side-by-side comparison may help as
before.

Dividing rational expressions follows the same procedure as dividing
rational numbers: we multiply the first term by the reciprocal of the
second. We finish by reducing the product to lowest terms.
4𝑦
÷
𝑦2
𝑥
=?
For struggling students, give

If 𝒂, 𝒃, 𝒄, and 𝒅 are rational expressions with 𝒃 ≠ 𝟎, 𝒄 ≠ 𝟎, and 𝒅 ≠ 𝟎, then
𝒂 𝒄 𝒂 𝒅 𝒂𝒅
÷ = ∙ =
.
𝒃 𝒅 𝒃 𝒄 𝒃𝒄
𝑥3
3 4 21
÷ =
5 7 20

𝑥2 𝑦
4
3𝑦 2
𝑧−1
÷
÷
𝑥𝑦 2
8
2𝑥
=
𝑦
12𝑦 5
(𝑧−1)2
=
(𝑧−1)
4𝑦 3
.
For advanced students, give


Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
𝑥−3
𝑥 2 +𝑥−2
÷
𝑥 2 −2𝑥−24
𝑥 2 −4
𝑥 2 −𝑥−6
𝑥−1
÷
=
1
(𝑥+2)2
𝑥 2 +3𝑥−4
𝑥 2 +𝑥−2
=
𝑥−6
.
𝑥−2
272
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Example 3 (3 minutes)
As in Example 2, ask students to apply their knowledge of rational-number division to rational expressions by working on
their own or with a partner. Circulate to assist and assess understanding. Once students have made attempts to divide,
use the scaffolded questions to develop the concept as necessary.
Example 3
Find the quotient and reduce to lowest terms:

First, we change the division of
𝑥 2 −4
3𝑥
𝒙𝟐 −𝟒
by
𝟑𝒙
÷
𝑥−2
2𝑥
𝒙−𝟐
𝟐𝒙
.
into multiplication of
𝑥 2 −4
3𝑥
by the multiplicative inverse of
𝑥−2
2𝑥
.
𝑥 2 − 4 𝑥 − 2 𝑥 2 − 4 2𝑥
÷
=
∙
3𝑥
2𝑥
3𝑥
𝑥−2

Then, we perform multiplication as in the previous examples and exercises. That is, we factor the numerator
and denominator and divide any common factors present in both the numerator and denominator.
𝑥 2 − 4 𝑥 − 2 (𝑥 − 2)(𝑥 + 2) 2𝑥
÷
=
∙
3𝑥
2𝑥
3𝑥
𝑥−2
2(𝑥 + 2)
=
3
Exercise 4 (3 minutes)
Allow students to work in pairs or small groups to evaluate the following quotient.
Exercises 4–5
4.
Find the quotient and reduce to lowest terms:
𝒙𝟐 −𝟓𝒙+𝟔
𝒙+𝟒
÷
𝒙𝟐 −𝟗
.
𝒙𝟐 +𝟓𝒙+𝟒
𝒙𝟐 − 𝟓𝒙 + 𝟔
𝒙𝟐 − 𝟗
𝒙𝟐 − 𝟓𝒙 + 𝟔 𝒙𝟐 + 𝟓𝒙 + 𝟒
÷ 𝟐
=
∙
𝒙+𝟒
𝒙 + 𝟓𝒙 + 𝟒
𝒙+𝟒
𝒙𝟐 − 𝟗
(𝒙 − 𝟑)(𝒙 − 𝟐) (𝒙 + 𝟒)(𝒙 + 𝟏)
=
∙
(𝒙 − 𝟑)(𝒙 + 𝟑)
𝒙+𝟒
=
(𝒙 − 𝟐)(𝒙 + 𝟏)
(𝒙 + 𝟑)
Discussion (4 minutes)
What do we do when the numerator and denominator of a fraction are themselves fractions? We call a fraction that
contains fractions a complex fraction. Remind students that the fraction bar represents division, so a complex fraction
represents division between rational expressions.
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
273
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Allow students the opportunity to simplify the following complex fraction.
12⁄
49
27⁄
28
Allow students to struggle with the problem before discussing solution methods.
12⁄
49 = 12 ÷ 27
27⁄
49 28
28
12 28
=
∙
49 27
3∙4 4∙7
=
∙
7 ∙ 7 33
2
4
=
7 ∙ 32
16
=
63
Notice that in simplifying the complex fraction above, we are merely performing division of rational numbers, and we
already know how to do that. Since we already know how to divide rational expressions, we can also simplify rational
expressions whose numerators and denominators are rational expressions.
Exercise 5 (4 minutes)
Allow students to work in pairs or small groups to simplify the following rational
expression.
5.
Simplify the rational expression.
𝒙+𝟐
)
𝒙𝟐 − 𝟐𝒙 − 𝟑
𝒙𝟐 − 𝒙 − 𝟔
( 𝟐
)
𝒙 + 𝟔𝒙 + 𝟓
(
𝒙+𝟐
𝒙+𝟐
𝒙𝟐 − 𝒙 − 𝟔
𝒙𝟐 − 𝟐𝒙 − 𝟑 =
÷ 𝟐
𝟐
𝟐
𝒙 −𝒙−𝟔
𝒙 − 𝟐𝒙 − 𝟑 𝒙 + 𝟔𝒙 + 𝟓
𝒙𝟐 + 𝟔𝒙 + 𝟓
𝒙+𝟐
𝒙𝟐 + 𝟔𝒙 + 𝟓
= 𝟐
∙ 𝟐
𝒙 − 𝟐𝒙 − 𝟑 𝒙 − 𝒙 − 𝟔
(𝒙 + 𝟓)(𝒙 + 𝟏)
𝒙+𝟐
=
∙
(𝒙 − 𝟑)(𝒙 + 𝟏) (𝒙 − 𝟑)(𝒙 + 𝟐)
𝒙+𝟓
=
(𝒙 − 𝟑)𝟐
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
Scaffolding:
For struggling students, give a
simpler example, such as
2𝑥
( )
2𝑥 6𝑥
3𝑦
=
÷ 2
6𝑥
( 2 ) 3𝑦 4𝑦
4𝑦
2𝑥 4𝑦 2
=
∙
3𝑦 6𝑥
4
= 𝑦
9
274
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 24
M1
ALGEBRA II
Closing (3 minutes)
Ask students to summarize the important parts of the lesson in writing, to a partner, or as a class. Use this as an
opportunity to informally assess understanding of the lesson. In particular, ask students to articulate the processes for
multiplying and dividing rational expressions and simplifying complex rational expressions either verbally or symbolically.
Lesson Summary
In this lesson, we extended multiplication and division of rational numbers to multiplication and division of rational
expressions.

To multiply two rational expressions, multiply the numerators together and multiply the denominators
together, and then reduce to lowest terms.

To divide one rational expression by another, multiply the first by the multiplicative inverse of the
second, and reduce to lowest terms.

To simplify a complex fraction, apply the process for dividing one rational expression by another.
Exit Ticket (4 minutes)
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Name
Date
Lesson 24: Multiplying and Dividing Rational Expressions
Exit Ticket
Perform the indicated operations, and reduce to lowest terms.
1.
𝑥−2
𝑥2 + 𝑥 − 2
∙
𝑥 2 − 3𝑥 + 2
𝑥+2
𝑥−2
2.
( 2
)
𝑥 +𝑥−2
𝑥2 − 3𝑥 + 2
)
𝑥+2
(
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Exit Ticket Sample Solutions
Perform the indicated operations, and reduce to lowest terms.
1.
𝒙−𝟐
𝒙𝟐 +𝒙−𝟐
∙
(𝒙 − 𝟏)(𝒙 − 𝟐)
𝒙−𝟐
𝒙𝟐 − 𝟑𝒙 + 𝟐
𝒙−𝟐
∙
=
∙
(𝒙 − 𝟏)(𝒙 + 𝟐)
𝒙𝟐 + 𝒙 − 𝟐
𝒙+𝟐
𝒙+𝟐
(𝒙 − 𝟐)𝟐
=
(𝒙 + 𝟐)𝟐
𝒙𝟐 −𝟑𝒙+𝟐
𝒙+𝟐
𝒙−𝟐
𝟐
)
𝒙𝟐 + 𝒙 − 𝟐 = 𝒙 − 𝟐 ÷ 𝒙 − 𝟑𝒙 + 𝟐
𝒙𝟐 − 𝟑𝒙 + 𝟐
𝒙𝟐 + 𝒙 − 𝟐
𝒙+𝟐
(
)
𝒙+𝟐
𝒙−𝟐
𝒙+𝟐
= 𝟐
∙
𝒙 + 𝒙 − 𝟐 𝒙𝟐 − 𝟑𝒙 + 𝟐
𝒙−𝟐
𝒙+𝟐
=
∙
(𝒙 − 𝟏)(𝒙 + 𝟐) (𝒙 − 𝟐)(𝒙 − 𝟏)
𝟏
=
(𝒙 − 𝟏)𝟐
𝒙−𝟐
2.
(
( 𝟐
)
𝒙 +𝒙−𝟐
𝒙𝟐 −𝟑𝒙+𝟐
)
𝒙+𝟐
(
Problem Set Sample Solutions
1.
Perform the following operations:
a.
Multiply
𝟏
𝟑
(𝒙 − 𝟐) by 𝟗.
b.
𝟑𝒙 − 𝟔
d.
Divide
𝟏
(𝒙 − 𝟖) by
𝟒
𝟏
.
c.
𝟏𝟐
𝟑𝒙 − 𝟐𝟒
𝟏 𝟐
(
𝟑 𝟓
𝒙−
𝟏
𝟓
𝟏
) by 𝟏𝟓.
𝟐𝒙 − 𝟏
2.
Divide
e.
Multiply
Multiply
𝟏 𝟏
(
𝟒 𝟑
𝒙 + 𝟐) by 𝟏𝟐.
𝒙+𝟔
𝟐
𝟑
𝟐
𝟗
(𝟐𝒙 + 𝟑) by 𝟒.
𝟑𝒙 + 𝟏
f.
Multiply 𝟎. 𝟎𝟑(𝟒 − 𝒙) by 𝟏𝟎𝟎.
𝟏𝟐 − 𝟑𝒙
Write each rational expression as an equivalent rational expression in lowest terms.
a.
𝒂𝟑 𝒃𝟐
𝒄
𝒂
(𝒄𝟐 𝒅𝟐 ∙ 𝒂𝒃) ÷ 𝒄𝟐 𝒅𝟑
b.
𝒂𝒃𝒄𝒅
d.
𝟑𝒙𝟐 −𝟔𝒙
𝟑𝒙+𝟏
𝒂𝟐 +𝟔𝒂+𝟗 𝟑𝒂−𝟗
𝒂𝟐 −𝟗
∙
𝒂+𝟑
c.
𝒙+𝟑𝒙𝟐
𝟑𝒙𝟐
𝒙−𝟐
Lesson 24:
e.
÷
𝟒𝒙
𝒙𝟐 −𝟏𝟔
𝟑(𝒙 + 𝟒)
𝟖
𝟑
∙ 𝒙𝟐 −𝟒𝒙+𝟒
𝟔𝒙
𝟒𝒙−𝟏𝟔
𝟐𝒙𝟐 −𝟏𝟎𝒙+𝟏𝟐 𝟐+𝒙
𝒙𝟐 −𝟒
∙ 𝟑−𝒙
−𝟐
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
f.
𝒂−𝟐𝒃
𝒂+𝟐𝒃
−
÷ (𝟒𝒃𝟐 − 𝒂𝟐 )
𝟏
(𝒂 + 𝟐𝒃)𝟐
277
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Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
g.
𝒅+𝒄
𝒄𝟐 +𝒅𝟐
−
j.
𝒄𝟐
÷
𝒄𝟐 −𝒅𝟐
h.
𝒅𝟐 −𝒅𝒄
𝒅
+ 𝒅𝟐
𝟗𝒂𝟐 −𝒃𝟐
÷
𝟏𝟔𝒂𝟐 −𝒃𝟐
i.
𝟑𝒂𝒃+𝒃𝟐
𝒃
𝟒𝒂 + 𝒃
𝒙−𝟐 −𝟑
(𝒙𝟐 +𝟏)
𝒙𝟐 −𝟒𝒙+𝟒
÷(
)
k.
𝒙𝟐 −𝟐𝒙−𝟑
(𝒙 − 𝟑)(𝒙 + 𝟏)(𝒙𝟐 + 𝟏)𝟑
(𝒙 − 𝟐)𝟓
3.
𝟏𝟐𝒂𝟐 −𝟕𝒂𝒃+𝒃𝟐
𝒙𝟐 −𝒙−𝟔
∙(
𝒙−𝟐
)
(𝒙 + 𝟐)𝟐
𝟔𝒙𝟐 −𝟏𝟏𝒙−𝟏𝟎
𝟔𝒙𝟐 −𝟓𝒙−𝟔
−
𝒙−𝟑 −𝟏
(𝒙𝟐 −𝟒)
𝟔−𝟒𝒙
∙ 𝟐𝟓−𝟐𝟎𝒙+𝟒𝒙𝟐
l.
𝟐
𝟐
, or
𝟐𝒙−𝟓
𝟓−𝟐𝒙
𝟑𝒙𝟑 −𝟑𝒂𝟐 𝒙
∙
𝒂−𝒙
𝒙𝟐 −𝟐𝒂𝒙+𝒂𝟐 𝒂𝟑 𝒙+𝒂𝟐 𝒙𝟐
−
𝟑
𝒂𝟐
Write each rational expression as an equivalent rational expression in lowest terms.
𝟒𝒂
)
𝟔𝒃𝟐
𝟐𝟎𝒂𝟑
(
a.
(
𝟏𝟐𝒃
𝟐
𝟓𝒂𝟐 𝒃
)
𝒙−𝟐
b.
( 𝟐 )
𝒙 −𝟏
𝒙−𝟔
(𝒙 + 𝟐)(𝒙𝟐 − 𝟏)
𝒙𝟐 −𝟒
(
)
𝒙−𝟔
𝒙𝟐 +𝟐𝒙−𝟑
c.
4.
( 𝟐
)
𝒙 +𝟑𝒙−𝟒
𝒙𝟐 +𝒙−𝟔
(
)
𝒙+𝟒
Suppose that 𝒙 =
𝟏
𝒙−𝟐
𝒕𝟐 +𝟑𝒕−𝟒
𝒕𝟐 +𝟐𝒕−𝟖
and 𝒚 = 𝟐
, for 𝒕 ≠ 𝟏, 𝒕 ≠ −𝟏, 𝒕 ≠ 𝟐, and 𝒕 ≠ −𝟒. Show that the value of
𝟐
𝟑𝒕 −𝟑
𝟐𝒕 −𝟐𝒕−𝟒
𝒙𝟐 𝒚−𝟐 does not depend on the value of 𝒕.
𝟐
−𝟐
𝒕𝟐 + 𝟑𝒕 − 𝟒
𝒕𝟐 + 𝟐𝒕 − 𝟖
𝒙𝟐 𝒚−𝟐 = (
) ( 𝟐
)
𝟑𝒕𝟐 − 𝟑
𝟐𝒕 − 𝟐𝒕 − 𝟒
𝟐
𝟐
𝒕𝟐 + 𝟑𝒕 − 𝟒
𝒕𝟐 + 𝟐𝒕 − 𝟖
=(
) ÷( 𝟐
)
𝟐
𝟑𝒕 − 𝟑
𝟐𝒕 − 𝟐𝒕 − 𝟒
𝟐
𝟐
𝒕𝟐 + 𝟑𝒕 − 𝟒
𝟐𝒕𝟐 − 𝟐𝒕 − 𝟒
=(
) ( 𝟐
)
𝟐
𝟑𝒕 − 𝟑
𝒕 + 𝟐𝒕 − 𝟖
𝟐
𝟐
(𝒕 − 𝟏)(𝒕 + 𝟒)
𝟐(𝒕 − 𝟐)(𝒕 + 𝟏)
=(
) (
)
(𝒕 − 𝟐)(𝒕 + 𝟒)
𝟑(𝒕 − 𝟏)(𝒕 + 𝟏)
=
𝟒(𝒕 − 𝟏)𝟐 (𝒕 + 𝟒)𝟐 (𝒕 − 𝟐)𝟐 (𝒕 + 𝟏)𝟐
𝟗(𝒕 − 𝟏)𝟐 (𝒕 + 𝟏)𝟐 (𝒕 − 𝟐)𝟐 (𝒕 + 𝟒)𝟐
=
𝟒
𝟗
𝟒
𝟗
Since 𝒙𝟐 𝒚−𝟐 = , the value of 𝒙𝟐 𝒚−𝟐 does not depend on 𝒕.
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
278
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 24
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
5.
Determine which of the following numbers is larger without using a calculator,
𝒂
𝒂
𝒃
𝒃
compare two positive quantities 𝒂 and 𝒃 by computing the quotient . If
𝒂
𝟎 < < 𝟏, then 𝒂 < 𝒃.)
𝒃
𝟏𝟓𝟏𝟔
𝟏𝟔𝟏𝟓
or
𝟐𝟎𝟐𝟒
𝟐𝟒𝟐𝟎
. (Hint: We can
> 𝟏, then 𝒂 > 𝒃. Likewise, if
𝟏𝟓𝟏𝟔 𝟐𝟎𝟐𝟒 𝟏𝟓𝟏𝟔 𝟐𝟒𝟐𝟎
÷
=
∙
𝟏𝟔𝟏𝟓 𝟐𝟒𝟐𝟎 𝟏𝟔𝟏𝟓 𝟐𝟎𝟐𝟒
=
𝟑𝟏𝟔 ∙ 𝟓𝟏𝟔 (𝟐𝟑 )𝟐𝟎 ∙ 𝟑𝟐𝟎
∙
(𝟐𝟒 )𝟏𝟓 (𝟐𝟐 )𝟐𝟒 ∙ 𝟓𝟐𝟒
=
𝟐𝟔𝟎 ∙ 𝟑𝟑𝟔 ∙ 𝟓𝟏𝟔
𝟐𝟏𝟎𝟖 ∙ 𝟓𝟐𝟒
=
=
𝟑𝟑𝟔
∙ 𝟓𝟖
𝟐𝟒𝟖
𝟗𝟏𝟖
∙ 𝟏𝟎𝟖
𝟐𝟒𝟎
𝟗 𝟖 𝟗𝟏𝟎
= ( ) ∙ ( 𝟒𝟎 )
𝟏𝟎
𝟐
𝟗 𝟖
𝟗 𝟏𝟎
=( ) ∙( )
𝟏𝟎
𝟏𝟔
Since
𝟏𝟓𝟏𝟔
𝟏𝟔𝟏𝟓
𝟗
𝟏𝟔
<
< 𝟏, and
𝟐𝟎𝟐𝟒
𝟗
𝟏𝟎
< 𝟏, we know that (
𝟏𝟓𝟏𝟔 𝟐𝟎𝟐𝟒
𝟗 𝟖
𝟗 𝟏𝟎
) ∙ ( ) < 𝟏. Thus, 𝟏𝟓 ÷ 𝟐𝟎 < 𝟏, and we know that
𝟏𝟎
𝟏𝟔
𝟏𝟔
𝟐𝟒
𝟐𝟎 .
𝟐𝟒
Extension:
6.
One of two numbers can be represented by the rational expression
a.
𝒙−𝟐
𝒙
, where 𝒙 ≠ 𝟎 and 𝒙 ≠ 𝟐.
Find a representation of the second number if the product of the two numbers is 𝟏.
Let the second number be 𝐲. Then (
𝒙−𝟐
) ∙ 𝒚 = 𝟏, so we have
𝒙
𝒙−𝟐
𝒚 =𝟏÷(
)
𝒙
𝒙
=𝟏∙(
)
𝒙−𝟐
𝒙
=
.
𝒙−𝟐
b.
Find a representation of the second number if the product of the two numbers is 𝟎.
Let the second number be 𝒛. Then (
𝒙−𝟐
) ∙ 𝒛 = 𝟎, so we have
𝒙
𝒙−𝟐
𝒛 =𝟎÷(
)
𝒙
𝒙
=𝟎∙(
)
𝒙−𝟐
= 𝟎.
Lesson 24:
Multiplying and Dividing Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
279
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.