2. Rational Expressions and Functions
Chapter 8
Rational Expressions and Functions

3. Adding and Subtracting Rational Expressions
8.2
Adding and Subtracting
Rational Expressions

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

5. 8.2 Adding and Subtracting 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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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

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

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

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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.
· m3 · n2
2 · 3 · m2 · n5
Choose the factors with
the greatest exponents.
LCD = · 3 · m3 · n5
= m3n5
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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 = (n – 5)(n – 3)
The LCD, divisible by both polynomials, is (n – 5)(n + 2)(n – 3).
Factor.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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
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).
Factor.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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

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

17. 8.2 Adding and Subtracting Rational Expressions
Caution with Subtracting Rational Expressions
CAUTION
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.
Subtract the numerators;
keep the common denominator.
d + 5
d – 6 8d – =
d + 5
8d – (d – 6) =
d + 5
8d – d + 6 =
d + 5
7d + 6
Combine terms in the numerator.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

18. 8.2 Adding and Subtracting Rational Expressions
EXAMPLE 4
Using the Distributive Property When
Subtracting Rational Expressions
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) –
Fundamental
property =
(e – 2)(e + 2)
2(e + 2) – 9(e – 2)
Subtract. =
(e – 2)(e + 2)
2e + 4 – 9e + 18
Distributive property =
(e – 2)(e + 2)
–7e + 22
Combine terms in
the numerator.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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

20. 8.2 Adding and Subtracting Rational Expressions
EXAMPLE 5
Adding Rational Expressions with
Denominators That Are Opposites
5 – x x
x – 5 4 + =
(5 – x)(–1)
x(–1)
x – 5 4 +
Add. =
x – 5 –x 4 +
Opposites =
x – 5
4 – x
Add the numerators.
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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

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
Factor each denominator.
The LCD is (a – 1)(a + 3)(a – 4). =
(a – 1)(a + 3)(a – 4)
(a – 2)(a – 4) –
(a – 1)(a – 4)(a + 3)
(a + 2)(a + 3)
Fundamental
property =
(a – 1)(a + 3)(a – 4)
(a – 2)(a – 4) – (a + 2)(a + 3)
Subtract. =
(a – 1)(a + 3)(a – 4)
a2 – 6a + 8 – (a2 + 5a + 6)
Multiply in the
numerator.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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)
Subtract. =
(a – 1)(a + 3)(a – 4)
a2 – 6a + 8 – (a2 + 5a + 6)
Multiply in the
numerator. =
(a – 1)(a + 3)(a – 4)
a2 – 6a + 8 – a2 – 5a – 6)
Distributive
property =
(a – 1)(a + 3)(a – 4)
–11a + 2
Combine terms in
the numerator.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

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
Factor each denominator.
The LCD is (b + 2)2(b + 1) =
(b + 2)2(b + 1)
2(b + 1) +
8(b + 2)
Fundamental
property =
(b + 2)2(b + 1)
2(b + 1) + 8(b + 2)
Add. =
(b + 2)2(b + 1)
2b b + 16
Distributive
property =
(b + 2)2(b + 1)
10b + 18
Combine like terms
in the numerator.
Copyright © 2010 Pearson Education, Inc. All rights reserved.