# 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.
Sec 8.2 - 1
2. Chapter 8
Rational Expressions and
Functions
Sec 8.2 - 2
3. 8.2
Rational Expressions
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.
5. 8.2 Adding and Subtracting Rational Expressions
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.
6. 8.2 Adding and Subtracting Rational Expressions
EXAMPLE 1 Adding and Subtracting Rational Expressions
with the Same Denominator
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
7. 8.2 Adding and Subtracting Rational Expressions
EXAMPLE 1 Adding and Subtracting Rational Expressions
with the Same Denominator
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
8. 8.2 Adding and Subtracting Rational Expressions
EXAMPLE 1 Adding and Subtracting Rational Expressions
with the Same Denominator
5 + k
k2 + 2k – 15 k2 + 2k – 15
=
k2 + 2k – 15
5+k
= Factor.
(k – 3)(k + 5)
1
= Lowest terms
k–3
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.
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
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. 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).
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).
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).
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
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.
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
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.
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.
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
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
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)
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.
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.
24. 8.2 Adding and Subtracting 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)