Adding and Subtracting Rational Expressions

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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.

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

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

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

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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 4

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 commo
denominator.
Step 2 Simplify. Write all answers in lowest terms.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 5

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 5n 4m + 5n Add the numerators.
(a) + =
7 7 7 Keep the common denominator.
1 – 5 1 – 5 Subtract the numerators; keep the
(b) =
g3 g3 g3 common denominator.
= – 4 Simplify.
g3
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 6

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 – b a–b Subtract the numerators; keep
(c) =
a2 – b2 a2 – b 2 a2 – b 2 the common denominator.
a–b
= Factor.
(a – b)(a + b)
1
= a+b Lowest terms
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 7

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 + k
k2 + 2k – 15 k2 + 2k – 15
5+k Add.
=
k2 + 2k – 15
5+k
= Factor.
(k – 3)(k + 5)
1
= Lowest terms
k–3
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 8

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 raise
to the greatest power that occurs in the denominator.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 9

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 · n 2
2 · 3 · m2 · n 5
Choose the factors with
LCD = 22 · 3 · m3 · n5 the greatest exponents.
= 12m3n5
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 10

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. Sec 8.2 - 11

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)
Factor.
n2 – 8n + 15 = (n – 5)(n – 3)
The LCD, divisible by both polynomials, is (n – 5)(n + 2)(n – 3).
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 12

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)
Factor.
3h – 9 = 3(h – 3)
The LCD is 4h·3·(h – 3) = 12h(h – 3).
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 13

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
Factor.
g2 + 3g – 4 = (g – 1)(g + 4)
5g + 20 = 5(g + 4)
The LCD is 5(g – 1)2(g + 4).
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 14

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.
(a) 7 1
+
3z 9z
7 1 = 7·3 1 Fundamental property
+ +
3z 9z 3z · 3 9z
= 21 1
+
9z 9z
= 21 + 1 Add the numerators.
9z
= 22
9z
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 15

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).
(b) 2 – 3 2(y – 4) – y·3 Fundamental property
=
y y–4 y(y – 4) y(y – 4)
2y – 8 – 3y Distributive and
=
y(y – 4) y(y – 4) commutative properties
2y – 8 – 3y Subtract the
=
y(y – 4) numerators.
–y – 8 Combine like terms in
=
y(y – 4) the numerator.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 16

17. 8.2 Adding and Subtracting Rational Expressions
Caution with Subtracting Rational Expressions
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.
8d – d–6 = 8d – (d – 6) = 8d – d + 6
d+5 d+5 d+5 d+5
= 7d + 6
Combine terms in the numerator.
d+5
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 17

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).
2 – 9 = 2(e + 2) – 9(e – 2) Fundamental
e–2 e+2 (e – 2)(e + 2) (e + 2)(e – 2) property
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
(e – 2)(e + 2) the numerator.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 8.2 - 18

19. 8.2 Adding and Subtracting Rational Expressions
EXAMPLE 5 Adding Rational Expressions with
Denominators That Are Opposites
Add. 4 + x = 4 + x(–1)
x–5 5–x x–5 (5 – x)(–1)
= 4 + –x
Opposites
x–5 x–5
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. Sec 8.2 - 19

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

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

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

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

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

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