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Chapter Chapter 77Section Section 44
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Adding and Subtracting Rational Expressions
Add rational expressions having the same denominator.Add rational expressions having different denominators.Subtract rational expressions.
11
33
22
7.47.47.47.4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Add rational expressions having the same denominator.
Slide 7.4 - 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Add rational expressions having the same denominator.
We find the sum of two rational expressions with the same procedure that we used in Section 1.1 for adding two fractions having the same denominator.
Slide 7.4 - 4
If and (Q ≠ 0) are rational expressions, then
That is, to add rational expressions with the same denominator, add the numerators and keep the same denominator.
.P R P R
Q Q Q
P
QR
Q
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Add. Write each answer in lowest terms.
Solution:
Adding Rational Expressions with the Same Denominator
Slide 7.4 - 5
7 3
15 15
2 2x y
x y x y
7 3
15
10
15
5
2
3
5
2
3
2 2x y
x y
2 x y
x y
2
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Add rational expressions having different denominators.
Slide 7.4 - 6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
We use the following steps, which are the same as those used in Section 1.1 to add fractions having different denominators.
Add rational expressions having different denominators.
Slide 7.4 - 7
Step 1: Find the least common denominator (LCD).
Step 2: Rewrite each rational expression as an equivalent rational expression with the LCD as the denominator.
Step 3: Add the numerators to get the numerator of the sum. The LCD is the denominator of the sum.
Step 4: Write in lowest terms using the fundamental property.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Add. Write each answer in lowest terms.
EXAMPLE 2
Solution:
Adding Rational Expressions with Different Denominators
Slide 7.4 - 8
1 1
10 15
2
3 7
m
n n
10 52
LCD 3 7 21n n
15 53 LCD 2 3 5 30
3 2
30 2
1 1
1 15
3 2
30 30
5
30
1
6
7 3
7
2
33 7
m
n n
7 6
21 21
m
n n
7 6
21
m
n
3 2
30
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Solution:
2
2 4
1 1
p
p p
Adding Rational Expressions
Slide 7.4 - 9
Add. Write the answer in lowest terms.
2 4
1 1 1
p
p p p
2 2
1 1
p
p p
2 4
1
1
11 1
p p
ppp p
2 2 4
1 1
p p
p p
12
11p
p
p
2
1p
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Solution:2 2
2 3
5 4 1
k
k k k
2 3
4 1 1 1
k
k k k k
Adding Rational Expressions
Add. Write the answer in lowest terms.
1 4
1
2 3
4 1 41 1
k
k k k
k
kk
k
k
2 1 3 4
4 1 1 4 1 1
k k k
k k k k k k
22 2 3 12
4 1 1
k k k
k k k
22 5 12
4 1 1
k k
k k k
2 3 4
4 1 1
k k
k k k
Slide 7.4 - 10
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5
Add. Write the answer in lowest terms.
Solution:
2 3 3 2
m n
m n n m
Adding Rational Expressions with Denominators That Are Opposites
Slide 7.4 - 11
2 3 3 2
1
1
m n
m n n m
2 3 3 2
m n
m n n m
2 3
m n
m n
2 3 2 3
m n
m n m n
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 33
Slide 7.4 - 12
Subtract rational expressions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
We subtract rational expressions having different denominators using a procedure similar to the one used to add rational expressions having different denominators.
Subtract rational expressions.
Slide 7.4 - 13
If and (Q ≠ 0) are rational expressions, then
That is, to subtract rational expressions with the same denominator, subtract the numerators and keep the same denominator.
P R P R
Q Q Q
R
Q
R
Q
Use the following rule to subtract rational expressions having the same denominator.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Subtract. Write the answer in lowest terms.
EXAMPLE 6
5 5
1 1
t t
t t
Subtracting Rational Expressions with the Same Denominator
Slide 7.4 - 14
5 5
1
t t
t
5 5
1
t t
t
4 5
1
t
t
Solution:
Sign errors often occur in subtraction problems. The numerator of the fraction being subtracted must be treated as a single quantity. Be sure to use parentheses after the subtraction sign.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7
6 1
2 3a a
Subtracting Rational Expressions with Different Denominators
Slide 7.4 - 15
Subtract. Write the answer in lowest terms.
Solution: 36 1
2 3
2
3 2
a a
a aa a
6 18 2
2 3 3 2
a a
a a a a
6 18 2
2 3
a a
a a
6 18 2
2 3
a a
a a
5 20
2 3
a
a a
5 4
2 3
a
a a
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 8
4 3 1
1 1
x x
x x
Subtracting Rational Expressions with Denominators That Are Opposite
Slide 7.4 - 16
Subtract. Write the answer in lowest terms.
Solution: 4 3 1
1 1
x x
x x
14 3 1
1 1 1
x x
x x
4 3 1
1
x x
x
4 3 1
1
x x
x
1
1
x
x
1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Subtract. Write the answer in lowest terms.
EXAMPLE 9
2 2
3 4
5 10 25
r
r r r r
Subtracting Rational Expressions
Slide 7.4 - 17
Solution:
3 4
5 5 5
r
r r r r
3 4
55 5
5
5
rr
r r
r
r rr r
23 15 4
5 5 5 5
r r r
r r r r r r
23 19
5 5
r r
r r r
3 19
5 5
r
r
r
r r
2
3 19
5
r
r
