Adding & Subtracting Like Rational Expressions

MATH 018

Combined Algebra

S. Rook

2

Overview

• Section 7.3 in the textbook:– Adding & subtracting rational expressions with

like denominators– Finding the LCD of two rational expressions– Writing equivalent rational expressions

Adding & Subtracting with Like Denominators

4

Adding & Subtracting with Like Denominators

• Consider adding 1/10 + 3/10

– Denominators are the same – add the numerators and keep the denominator

4/10

– Simplify the final answer if possible2/5

5

Adding & Subtracting with Like Denominators (Continued)

• Similar for rational expressions with equal denominators– Add the polynomial numerators and keep the

polynomial denominator– Simplify the final answer if possible

Adding & Subtracting with Like Denominators (Example)

Ex 1: Add/Subtract and simplify:

a)

b)

6

7

12

7

2

x

x

x

x

1

42

1

35

x

x

x

x

Finding the LCD of Two Rational Expressions

8

LCD of Rational Expressions

• Least Common Denominator (LCD/LCM): the smallest product of unique factors that each denominator will divide into evenly

• Consider 4, 12, and 16– What is the smallest number that all three will

divide into evenly?

• Consider x2(x + 1) and x(x – 1)– What is the smallest product that both

expressions will divide into evenly?

GCF vs LCM/LCD

• The common mistake is to mix up GCF (the largest possible product that divides evenly into each factor) and the LCM (the smallest possible product that each factor will divide into evenly)

• Consider 4x3(x + 2) and 6x– What is the GCF? What is the LCM?

• Consider 2(x – 3)2 and (x – 3)(x + 3)– What is the GCF? What is the LCM?

• Always attempt to factor first– Object is to find the simplest LCM/LCD possible

9

LCD of Rational Expressions (Example)

Ex 2: Find the LCD of the rational expressions:

a) d)

b) e)

c)10

1

3

1

4

xx

4

2

6

922

xxx

x

3

1

962

xxx

x

20

19

16 22

2

xx

x

x

x

844

2

862

922

xxxx

x

Writing Equivalent Rational Expressions

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Writing Equivalent Rational Expressions

• How can we write 2/3 with a denominator of 6?

• Same process for rational expressions

• How can we write 1/x2 with a denominator of x5?

Writing Equivalent Rational Expressions (Example)

Ex 3: Write an equivalent rational expression with the given denominator:

a)

b)

c)13

41

?

1

5

xxx

25?

5

12

xx

x

194

?

94

2

xxxxx

x

14

Summary

• After studying these slides, you should know how to do the following:– Add or subtract rational expressions with a common

denominator– Determine the LCD from two rational expressions– Write an equivalent rational expression over a

different denominator• Additional Practice

– See the list of suggested problems for 7.3• Next lesson

– Adding & Subtracting Rational Expressions with Unlike Denominators (Section 7.4)