Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Lesson 10: The Structure of Rational Expressions
Classwork
Opening Exercise
3
2
+ .
a.
Add the fractions:
b.
Subtract the fractions:
c.
Add the expressions:
d.
Subtract the expressions:
5
Lesson 10:
7
3
𝑥
5
2
4
− .
3
𝑥
+ .
5
𝑥
𝑥+2
−
3
.
𝑥+1
The Structure of Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M3-TE-1.3.0-08.2015
S.56
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10
M3
PRECALCULUS AND ADVANCED TOPICS
Exercises
1.
Construct an argument that shows that the set of rational numbers is closed under addition. That is, if 𝑥 and 𝑦 are
rational numbers and 𝑤 = 𝑥 + 𝑦, prove that 𝑤 must also be a rational number.
2.
How could you modify your argument to show that the set of rational numbers is also closed under subtraction?
Discuss your response with another student.
3.
Multiply the fractions:
2 3
∙ .
5 4
Lesson 10:
The Structure of Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M3-TE-1.3.0-08.2015
S.57
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10
M3
PRECALCULUS AND ADVANCED TOPICS
2
3
÷ .
4.
Divide the fractions:
5.
Multiply the expressions:
6.
Divide the expressions:
5
4
𝑥+1
∙
3𝑥
.
𝑥+2 𝑥−4
𝑥+1
𝑥+2
Lesson 10:
÷
3𝑥
.
𝑥−4
The Structure of Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M3-TE-1.3.0-08.2015
S.58
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10
M3
PRECALCULUS AND ADVANCED TOPICS
7.
Construct an argument that shows that the set of rational numbers is closed under division. That is, if 𝑥 and 𝑦 are
𝑥
rational numbers (with 𝑦 nonzero) and 𝑤 = , prove that 𝑤 must also be a rational number.
𝑦
8.
How could you modify your argument to show that the set of rational expressions is also closed under division by a
nonzero rational expression? Discuss your response with another student.
Lesson 10:
The Structure of Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M3-TE-1.3.0-08.2015
S.59
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
PRECALCULUS AND ADVANCED TOPICS
Problem Set
1.
2.
Given
𝑥+1
𝑥−2
and
𝑥−1
, show that performing the following operations results in another rational expression.
𝑥2 − 4
a.
Addition
b.
Subtraction
c.
Multiplication
d.
Division
Find two rational expressions
𝑎
𝑏
and
𝑐
𝑑
that produce the result
𝑥−1
𝑥2
when using the following operations. Answers
for each type of operation may vary. Justify your answers.
3.
a.
Addition
b.
Subtraction
c.
Multiplication
d.
Division
Find two rational expressions
𝑎
𝑏
and
𝑐
𝑑
that produce the result
2𝑥 + 2
𝑥 2 − 𝑥
when using the following operations. Answers
for each type of operation may vary. Justify your answers.
4.
a.
Addition
b.
Subtraction
c.
Multiplication
d.
Division
𝐴
Consider the rational expressions 𝐴, 𝐵 and their quotient, , where 𝐵 is not equal to zero.
𝐵
a.
b.
c.
For some rational expression 𝐶, does
𝐴𝐶
𝐵𝐶
𝐴
= ?
𝐵
𝑥 1
𝑦 1
Let 𝐴 = + and 𝐵 = + . What is the least common denominator of every term of each expression?
𝑦 𝑥
𝑥 𝑦
𝐴𝐶
Find 𝐴𝐶, 𝐵𝐶 where 𝐶 is equal to your result in part (b). Then, find
𝐵𝐶
. Simplify your answer.
d.
Express each rational expression 𝐴, 𝐵 as a single rational term, that is, as a division between two polynomials.
e.
Write
f.
Use your answers to parts (d) and (e) to simplify .
g.
Summarize your findings. Which method do you prefer using to simplify rational expressions?
𝐴
𝐵
as a multiplication problem.
𝐴
𝐵
Lesson 10:
The Structure of Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M3-TE-1.3.0-08.2015
S.60
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10
M3
PRECALCULUS AND ADVANCED TOPICS
5.
6.
Simplify the following rational expressions.
a.
1 1
−
𝑦 𝑥
𝑥 𝑦
−
𝑦 𝑥
b.
1 1
+
𝑥 𝑦
1
1
−
𝑥2 𝑦2
c.
1
1
−
𝑥4 𝑦2
1
2
1
+
+
𝑥4 𝑥2 𝑦 𝑦2
d.
1
1
−
𝑥−1 𝑥
1
1
+
𝑥−1 𝑥
Find 𝐴 and 𝐵 that make the equation true. Verify your results.
a.
b.
7.
𝐴
𝑥+1
𝐴
𝑥+3
+
+
𝐵
𝑥−1
𝐵
𝑥+2
=
=
2
𝑥2 − 1
2𝑥 − 1
𝑥 2 + 5𝑥 + 6
Find 𝐴, 𝐵, and 𝐶 that make the equation true. Verify your result.
𝐴𝑥 + 𝐵
𝐶
𝑥−1
+
=
𝑥 2 + 1 𝑥 + 2 (𝑥 2 + 1)(𝑥 + 2)
Lesson 10:
The Structure of Rational Expressions
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M3-TE-1.3.0-08.2015
S.61
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.