9.5 Adding and Subtracting Rational Expressions
4/23/2014
Same Denominator:
When adding or subtracting fractions with same denominators, add/subtract the numerators and
keep the same denominator
Example 1 Add and Subtract with Like Denominators
Perform the indicated operation.
2x3
2x5
+
Add numerators and simplify. 2x
32x5
+ =3 5+
2x=
2x8 ΒΏ
4 β 22 βπ₯ ΒΏ
ππ
Different Denominators:
If the denominators are different, find the least common denominator (LCD).
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:
Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
b. 12x 4β
13x 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)
Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
c. 3 ,x 2β
xx 2 4β
Example 2 Find the LCD of Rational Expressions
Find the least common denominator.
Factors
LCD:
x 1+x 1β,x
2x 1+d.
Basic Example of Adding Fractions with different denominators.
31
43
4: 4, 8, 12, 16..3: 3, 6, 9, 12β¦LCD: 12
33 β β 4
4
+ΒΏ12
= β12
ΒΏ9 4 13
Example 3 Add with Unlike Denominators
Perform the indicated operation.
a. 2x3 + x 2
4
Find LCD: 2 π₯2 β π₯π₯ β 22
3 π₯+8
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
ΒΏ2π₯2+13 π₯β6(π₯β2)(π₯+5)
π . 2π₯π₯β 2+
3π₯+5
Checkpoint Add and Subtract with Unlike Denominators
Perform the indicated operation.
2x37.
x 2
2+
ANSWER
2x 2+3x 4
Checkpoint Add and Subtract with Unlike Denominators
8. 3x 1β
xx 1++
ANSWER
x 2 + 2x + 3( )x β 1 ( )x + 1
Perform the indicated operation.
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:
9.5 p.497 #10, 12, 13, 15, 16-24even, 28, 30, 33, 37, 39
Homework:
Checkpoint Add and Subtract with Like Denominators
Perform the indicated operation.
1. 3x2
3x4
+ ANSWERx2
22. x 2β
xx 2β
β ANSWER β1
x2x
4+3.
x 4+β x 3β
ANSWERx 4+x 3+
Subtracting PolynomialsSwitch the signs of everything thatβs being subtracted and combine like terms.Ex:
Example 1 Subtract with Like Denominators
Subtract.
xx
3+ x2
3+β
Subtract numerators. x
x3+ x
23+
β =x 3+x 2β
Checkpoint Extra Examples
21. x 2β
xx 2β
β
ANSWER
β1
x2x
4+2.
x 4+β x 3β
x 4+x 3+
Example 2 Subtract with Unlike Denominators
Subtract .2x 2 ββ1
x 2
β x β 2 4
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.
Example 2 Subtract with Unlike Denominators
Subtract .2x 2 ββ1
x 2
β x β 2 4
SOLUTIONThe 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
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 + 2β
( )x + 1
2x 2 ββ1
x 2
β x β 2 4
= 1 2β Factor denominators.( )x β 2 ( )x + 1 ( )x β 2 ( )x + 2
9.5 p.497 # 11, 14, 29, 31, 32, 38,40, 41
Homework: