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9.5 Adding and Subtracting Rational Expressions
4/23/2014
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Same Denominator:
When adding or subtracting fractions with same denominators, add/subtract the numerators and
keep the same denominator
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Example 1 Add and Subtract with Like Denominators
Perform the indicated operation.
2x
3
2x
5+
Add numerators and simplify. 2x
3
2x
5+ =
3 5+
2x=
2x
8 ΒΏ4 β 22 βπ₯ ΒΏ
ππ
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Different Denominators:
If the denominators are different, find the least common denominator (LCD).
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Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
a.6x1 ,
8x 3
1
List the multiples of the number:6: 6, 12, 18, 24, 30, 36β¦.8: 8, 16, 24, 32β¦Take the Least Common Multiple (LCM): 24As for the variable, take the one with highest exponent: LCD:
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Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
b.1
2x 4β1
3x 9+,
2x - 4 = 2(x β 2)3x + 9 = 3(x + 3)2: 2, 4, 6, 8β¦3: 3, 6, 9, 12β¦LCD: 6(x-2)(x+3) (Take the LCM and multiply by the rest of their factors)
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Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
c.3 ,
x 2βx
x 2 4β
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Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
x 1+x 1β,x
2x 1+d.
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Basic Example of Adding Fractions with different denominators.
3
1
4
3 4: 4, 8, 12, 16..
3: 3, 6, 9, 12β¦LCD: 12
33
β β44
+ΒΏ12
= β12
ΒΏ9 4 13
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Example 3 Add with Unlike Denominators
Perform the indicated operation.
a.2x3
+x 2
4
Find LCD: 2 π₯2 βπ₯π₯
β22
3 π₯+8
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Example 3 Add with Unlike Denominators
Find LCD: (π₯β 2)(π₯+5) β
(π₯+5)(π₯+5) β
(π₯β2)(π₯β2)
2 π₯2+10 π₯+3 π₯β6
ΒΏ 2π₯2+13 π₯β6(π₯β2)(π₯+5)
π .2π₯π₯β 2
+3
π₯+5
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Checkpoint Add and Subtract with Unlike Denominators
Perform the indicated operation.
2x3
7.x 2
2+
ANSWER
2x 2
+3x 4
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Checkpoint Add and Subtract with Unlike Denominators
8.3
x 1βx
x 1++
ANSWER
x 2 + 2x + 3( )x β 1 ( )x + 1
Perform the indicated operation.
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Example 4 Add with Unlike Denominators
Find LCD: (π₯+6)(π₯β 6)
β(π₯β6)(π₯β6)
2 π₯+3 π₯β 18
ΒΏ5 π₯β18
(π₯β 6)(π₯+6)
Add:
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9.5 p.497 #10, 12, 13, 15, 16-24even, 28, 30, 33, 37, 39
Homework:
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Checkpoint Add and Subtract with Like Denominators
Perform the indicated operation.
1. 3x
2
3x
4+ ANSWER
x2
22.
x 2β
x
x 2ββ ANSWER β1
x
2x
4+3.
x 4+β
x 3βANSWER
x 4+
x 3+
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Subtracting PolynomialsSwitch the signs of everything thatβs being subtracted and combine like terms.
Ex:
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Example 1 Subtract with Like Denominators
Subtract.
x
x
3+ x
2
3+β
Subtract numerators. x
x
3+ x
2
3+β =
x 3+
x 2β
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Checkpoint Extra Examples
21.
x 2β
x
x 2ββ
ANSWER
β1
x
2x
4+2.
x 4+β
x 3β
x 4+
x 3+
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Example 2 Subtract with Unlike Denominators
Subtract .2x 2 β
β1
x 2
β x β 2 4
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Checkpoint Add and Subtract with Unlike Denominators
1 1βx 2
β 1 x + 1
ANSWER
β x β 2( )x β 1( )x + 1
Perform the indicated operation.
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Example 2 Subtract with Unlike Denominators
Subtract .2x 2 β
β1
x 2
β x β 2 4
SOLUTION
The factors of are and and the factors of are and . So, the LCD is .
x 2
β x β 2 ( )x β 2 ( )x + 1x 2 β 4 ( )x β 2 ( )x + 2
( )x + 2( )x β 2 ( )x + 1
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Example 2 Subtract with Unlike Denominators
Rewrite using LCD.
= β 2( )x + 2( )x β 2 ( )x + 1 ( )x β 2( )x + 2 ( )x + 2
( )x + 1
=Subtract numerators.
x + 2 β 2( )x + 1( )x β 2 ( )x + 1 ( )x + 2
= Distributive property
x + 2 β 2x( )x β 2 ( )x + 1 ( )x + 2
β 2
Combine like terms.=
x( )x β 2 ( )x + 1 ( )x + 2
β
( )x + 1
2x 2 β
β1
x 2
β x β 2 4
=1 2
β Factor denominators.( )x β 2 ( )x + 1 ( )x β 2 ( )x + 2
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9.5 p.497 # 11, 14, 29, 31, 32, 38,40, 41
Homework: