NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25
M1
ALGEBRA II
Lesson 25: Adding and Subtracting Rational Expressions
Classwork
Exercises 1–4
1.
2.
3.
4.
Calculate the following sum:
3
20
𝜋
4
𝑎
𝑚
−
+
+
3
10
+
6
10
.
4
15
√2
5
𝑏
2𝑚
−
𝑐
𝑚
Lesson 25:
Adding and Subtracting 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
S.124
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25
M1
ALGEBRA II
Example 1
Perform the indicated operations below and simplify.
a.
b.
c.
𝑎+𝑏
4
4
3𝑥
+
−
2𝑎−𝑏
5
3
5𝑥 2
3
2𝑥 2 +2𝑥
+
5
𝑥 2 −3𝑥−4
Lesson 25:
Adding and Subtracting 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
S.125
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 25
M1
ALGEBRA II
Exercises 5–8
Perform the indicated operations for each problem below.
5.
6.
7.
8.
5
𝑥−2
+
7𝑚
𝑚−3
3𝑥
4𝑥−8
+
5𝑚
3−𝑚
𝑏2
𝑏2 −2𝑏𝑐+𝑐 2
𝑥
𝑥 2 −1
−
−
𝑏
𝑏−𝑐
2𝑥
𝑥 2 +𝑥−2
Lesson 25:
Adding and Subtracting 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
S.126
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Example 2
Simplify the following expression.
2
𝑏 +𝑏−1−1
2𝑏 − 1
4− 8
(𝑏 + 1)
Lesson 25:
Adding and Subtracting 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
S.127
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
M1
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
ALGEBRA II
Lesson Summary
In this lesson, we extended addition and subtraction of rational numbers to addition and subtraction of rational
expressions. The process for adding or subtracting rational expressions can be summarized as follows:

Find a common multiple of the denominators to use as a common denominator.

Find equivalent rational expressions for each expression using the common denominator.

Add or subtract the numerators as indicated and simplify if needed.
Problem Set
1.
Write each sum or difference as a single rational expression.
a.
b.
c.
2.
8
√5
10
4
𝑥
√3
−
5
+
√2
6
+2
3
+
2𝑥
Write as a single rational expression.
a.
d.
g.
j.
m.
3.
7
1
𝑥
1
−
1
𝑝−2
1−
b.
𝑥−1
−
1
e.
𝑝+2
1
1+𝑝
3
𝑥−4
+
h.
2
k.
4−𝑥
1
2𝑚−4𝑛
−
1
2𝑚+4𝑛
−
𝑚
𝑚2 −4𝑛2
n.
3𝑥
2𝑦
−
1
𝑝−2
𝑝+𝑞
𝑝−𝑞
3𝑛
𝑛−2
5𝑥
6𝑦
+
+
𝑥
c.
3𝑦
1
f.
2−𝑝
i.
−2
+
3
l.
2−𝑛
1
(2𝑎−𝑏)(𝑎−𝑐)
+
1
o.
(𝑏−𝑐)(𝑏−2𝑎)
𝑎−𝑏
+
𝑎2
1
−
𝑏+1
𝑟
𝑠−𝑟
+
8𝑥
3𝑦−2𝑥
𝑏2 +1
𝑏2 −4
1
𝑎
𝑏
1+𝑏
𝑠
𝑟+𝑠
+
+
12𝑦
2𝑥−3𝑦
1
𝑏+2
+
1
𝑏−2
Write each rational expression as an equivalent rational expression in lowest terms.
a.
1
1
−
𝑎 2𝑎
4
𝑎
b.
Lesson 25:
5𝑥
+1
2
5𝑥
1
−
4
5𝑥
Adding and Subtracting 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
4𝑥 + 3
c.
1+ 2
𝑥 +1
𝑥+7
1− 2
𝑥 +1
S.128
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 25
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Extension:
4.
Suppose that 𝑥 ≠ 0 and 𝑦 ≠ 0. We know from our work in this section that
that
1
𝑥
+
1
𝑦
is equivalent to
1
𝑥+𝑦
1 1
∙
𝑥 𝑦
is equivalent to
1
𝑥𝑦
. Is it also true
? Provide evidence to support your answer.
2𝑡
1−𝑡2
and 𝑦 =
. Show that the value of 𝑥2 + 𝑦2 does not depend on the value of 𝑡.
2
1+𝑡
1+𝑡2
5.
Suppose that 𝑥 =
6.
Show that for any real numbers 𝑎 and 𝑏, and any integers 𝑥 and 𝑦 so that 𝑥 ≠ 0, 𝑦 ≠ 0, 𝑥 ≠ 𝑦, and 𝑥 ≠ −𝑦,
𝑦 𝑥 𝑎𝑥+𝑏𝑦 𝑎𝑥−𝑏𝑦
𝑥 − 𝑦) ( 𝑥+𝑦 − 𝑥−𝑦 ) = 2(𝑎 − 𝑏).
(
7.
Suppose that 𝑛 is a positive integer.
𝑃
1
for polynomials 𝑃 and 𝑄: (1 + 1
𝑛) (1 + 𝑛+1).
a.
Rewrite the product in the form
b.
Rewrite the product in the form
c.
Rewrite the product in the form
d.
If this pattern continues, what is the product of 𝑛 of these factors?
Lesson 25:
𝑄
𝑃
𝑄
𝑃
𝑄
1
1
for polynomials 𝑃 and 𝑄: (1 + 1
𝑛) (1 + 𝑛+1) (1 + 𝑛+2).
1
1
1
for polynomials 𝑃 and 𝑄: (1 + 1
𝑛) (1 + 𝑛+1) (1 + 𝑛+2) (1 + 𝑛+3).
Adding and Subtracting 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
S.129
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.