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1. Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 8.2 - 1

2. Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 8.2 - 2
Rational Expressions and Functions
Chapter 8

3. Copyright © 2010 Pearson Education, Inc. All rights reserved
Sec 8.2 - 3
8.2
Adding and Subtracting
Rational Expressions

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 4
8.2 Adding and Subtracting Rational Expressions
Objectives
1. Add and subtract rational expressions with the
same denominator.
2. Find a least common denominator.
3. Add and subtract rational expressions with
different denominators.

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 5
8.2 Adding and Subtracting Rational Expressions
Rational Expressions
Adding or Subtracting Rational Expressions
Step 1 If the denominators are the same, add or subtract the numerators.
Place the result over the common denominator.
If the denominators are different, first find the least common
denominator. Write all rational expressions with this LCD, and then
add or subtract the numerators. Place the result over the common
denominator.
Step 2 Simplify. Write all answers in lowest terms.

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 6
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 1 Adding and Subtracting Rational Expressions
with the Same Denominator
Add or subtract as indicated.
(a)
4m
7
5n
7
+
4m + 5n
7
=
Add the numerators.
Keep the common denominator.
(b)
1
g3
5
g3
– 1 – 5
g3
=
Subtract the numerators; keep the
common denominator.
4
g3
= – Simplify.

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 7
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 1 Adding and Subtracting Rational Expressions
with the Same Denominator
Add or subtract as indicated.
(c)
a
a2 – b2
b
a2 – b2
– a – b
a2 – b2
=
a – b
(a – b)(a + b)
=
1
a + b
=
Subtract the numerators; keep
the common denominator.
Factor.
Lowest terms

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 8
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 1 Adding and Subtracting Rational Expressions
with the Same Denominator
Add or subtract as indicated.
(d)
Add.
Factor.
Lowest terms
5
k2 + 2k – 15
+
k
k2 + 2k – 15
=
5 + k
k2 + 2k – 15
=
5 + k
(k – 3)(k + 5)
=
1
k – 3

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 9
8.2 Adding and Subtracting Rational Expressions
The Least Common Denominator
Finding the Least Common Denominator
Step 1 Factor each denominator.
Step 2 Find the least common denominator. The LCD is the product of
all different factors from each denominator, with each factor raised
to the greatest power that occurs in the denominator.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 10
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 2 Finding Least Common Denominators
Assume that the given expressions are denominators of fractions. Find the
LCD for each group.
(a) 4m3n2, 6m2n5
Factor each denominator.
22 · m3 · n2
2 · 3 · m2 · n5
LCD = 22 · 3 · m3 · n5
= 12m3n5
Choose the factors with
the greatest exponents.

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 11
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 2 Finding Least Common Denominators
Assume that the given expressions are denominators of fractions. Find the
LCD for each group.
(b) y – 5, y
Each denominator is already factored. The LCD, an expression
divisible by both y – 5 and y is
y(y – 5).
It is usually best to leave a least common denominator in factored form.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 12
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 2 Finding Least Common Denominators
Assume that the given expressions are denominators of fractions. Find the
LCD for each group.
(c) n2 – 3n – 10, n2 – 8n + 15
Factor the denominators.
n2 – 3n – 10 = (n – 5)(n + 2)
n2 – 8n + 15 = (n – 5)(n – 3)
The LCD, divisible by both polynomials, is (n – 5)(n + 2)(n – 3).
Factor.

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 13
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 2 Finding Least Common Denominators
Assume that the given expressions are denominators of fractions. Find the
LCD for each group.
(d) 4h2 – 12h, 3h – 9
4h2 – 12h = 4h(h – 3)
3h – 9 = 3(h – 3)
The LCD is 4h·3·(h – 3) = 12h(h – 3).
Factor.

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 14
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 2 Finding Least Common Denominators
Assume that the given expressions are denominators of fractions. Find the
LCD for each group.
(e) g2 – 2g + 1, g2 + 3g – 4, 5g + 20
Factor.
g2 – 2g + 1 = (g – 1)2
g2 + 3g – 4 = (g – 1)(g + 4)
5g + 20 = 5(g + 4)
The LCD is 5(g – 1)2(g + 4).

15. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 15
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 3 Adding and Subtracting Rational
Expressions with Different Denominators
Add or subtract as indicated.
(a)
3z
7
9z
1
+
3z
7
9z
1
+ =
3z · 3
7 · 3
9z
1
+
=
9z
21
9z
1
+
=
9z
22
Fundamental property
Add the numerators.
=
9z
21 + 1
The LCD of 3z and 9z is 9z.

16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 16
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 3 Adding and Subtracting Rational
Expressions with Different Denominators
Add or subtract as indicated.
(b)
y
2
y – 4
3
– Fundamental property
Distributive and
commutative properties
=
y(y – 4)
2(y – 4)
y(y – 4)
y · 3
–
=
y(y – 4)
2y – 8 – 3y
=
y(y – 4)
2y – 8
y(y – 4)
3y
–
=
y(y – 4)
–y – 8
Subtract the
numerators.
Combine like terms in
the numerator.
The LCD is y(y – 4).

17. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 17
Subtract the numerators;
keep the common denominator.
8.2 Adding and Subtracting Rational Expressions
Caution with Subtracting Rational Expressions
CAUTION
d + 5
d – 6
d + 5
8d
– =
d + 5
8d – (d – 6)
=
d + 5
8d – d + 6
One of the most common sign errors in algebra occurs when a rational
expression with two or more terms in the numerator is being subtracted.
In this situation, the subtraction sign must be distributed to every term
in the numerator of the fraction that follows it. Carefully study the
example below to see how this is done.
=
d + 5
7d + 6
Combine terms in the numerator.

18. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 18
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 4 Using the Distributive Property When
Subtracting Rational Expressions
Subtract.
Add or subtract as indicated. The LCD of (e – 2) and (e + 2) is (e – 2)(e + 2).
e + 2
9
e – 2
2
– =
(e – 2)(e + 2)
2(e + 2)
(e + 2)(e – 2)
9(e – 2)
–
=
(e – 2)(e + 2)
2(e + 2) – 9(e – 2)
Distributive property
=
(e – 2)(e + 2)
2e + 4 – 9e + 18
=
(e – 2)(e + 2)
–7e + 22 Combine terms in
the numerator.
Fundamental
property

19. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 19
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 5 Adding Rational Expressions with
Denominators That Are Opposites
Add.
5 – x
x
x – 5
4
+
To get a common denominator of x – 5, multiply the second
expression by –1 in both the numerator and the denominator.
=
(5 – x)(–1)
x(–1)
x – 5
4
+
=
x – 5
–x
x – 5
4
+
Opposites

20. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 20
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 5 Adding Rational Expressions with
Denominators That Are Opposites
Add.
5 – x
x
x – 5
4
+
Add the numerators.
=
(5 – x)(–1)
x(–1)
x – 5
4
+
=
x – 5
–x
x – 5
4
+
=
x – 5
4 – x
Opposites
If we had used 5 – x as the common denominator and rewritten
the first expression, we would have obtained
,
5 – x
x – 4
an equivalent answer. Verify this.

21. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 21
The denominator of the second rational expression factors as
n(n + 3), which is the LCD for the three rational expressions.
Fundamental property
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 6 Adding and Subtracting Three Rational
Expressions
Add.
Add the numerators.
n2 + 3n
2
n + 3
3
+
n
4
+ =
n(n + 3)
2
n(n + 3)
3n
+
n(n + 3)
4(n + 3)
+
=
n(n + 3)
3n + 2 + 4(n + 3)
=
n(n + 3)
3n + 2 + 4n + 12
=
n(n + 3)
7n + 14
=
n(n + 3)
7(n + 2)
Distributive property
Combine terms.
Factor the numerator.

22. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 22
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 7 Subtracting Rational Expressions
Add.
a2 + 2a – 3
a – 2
–
a2 – 5a + 4
a + 2
=
(a – 1)(a + 3)
a – 2
–
(a – 1)(a – 4)
a + 2
=
(a – 1)(a + 3)(a – 4)
(a – 2)(a – 4)
–
(a – 1)(a – 4)(a + 3)
(a + 2)(a + 3)
=
(a – 1)(a + 3)(a – 4)
(a – 2)(a – 4) – (a + 2)(a + 3)
=
(a – 1)(a + 3)(a – 4)
a2 – 6a + 8 – (a2 + 5a + 6)
Fundamental
property
Subtract.
Multiply in the
numerator.
The LCD is (a – 1)(a + 3)(a – 4).
Factor each denominator.

23. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 23
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 7 Subtracting Rational Expressions
Add.
a2 + 2a – 3
a – 2
–
a2 – 5a + 4
a + 2
=
(a – 1)(a + 3)
a – 2
–
(a – 1)(a – 4)
a + 2
=
(a – 1)(a + 3)(a – 4)
(a – 2)(a – 4)
–
(a – 1)(a – 4)(a + 3)
(a + 2)(a + 3)
=
(a – 1)(a + 3)(a – 4)
(a – 2)(a – 4) – (a + 2)(a + 3)
=
(a – 1)(a + 3)(a – 4)
a2 – 6a + 8 – (a2 + 5a + 6)
=
(a – 1)(a + 3)(a – 4)
a2 – 6a + 8 – a2 – 5a – 6)
=
(a – 1)(a + 3)(a – 4)
–11a + 2
Subtract.
Multiply in the
numerator.
Distributive
property
Combine terms in
the numerator.

24. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 24
8.2 Adding and Subtracting Rational Expressions
EXAMPLE 8 Adding Rational Expressions
Add.
b2 + 4b + 4
2 +
b2 + 3a + 2
8
=
(b + 2)2
2 +
(b + 1)(b + 2)
8
=
(b + 2)2(b + 1)
2(b + 1)
+
(b + 2)2(b + 1)
8(b + 2)
=
(b + 2)2(b + 1)
2(b + 1) + 8(b + 2)
=
(b + 2)2(b + 1)
2b + 2 + 8b + 16
=
(b + 2)2(b + 1)
10b + 18
Fundamental
property
Add.
Distributive
property
Combine like terms
in the numerator.
The LCD is (b + 2)2(b + 1)
Factor each denominator.