6-3: complex rational expressions complex rational expression (fraction) – contains a fraction in...
TRANSCRIPT
6-3: Complex Rational Expressions
14
x
x 2
2
35
8
10 3
9
xx
y
x x
y
1 sincos cos1 sincos cos
complex rational expression (fraction) – contains a fraction in its numerator, denominator, or both.
Method 1: Find LCD of all denominators, then multiply both the numerator and denominator by the LCD.
17
15
y
y
LCD = y
17
15
y
y
Distribute 7
5
yy
yy
yy
Ex 1:
1 7
1 5
y
y
Multiply numerator and denominator by LCD
1
y
1
y
Ex 2: LCD =
1
1
yy
yy
y
1
y
1
yDistribute
2
2
yy
yy
yy
2
21
1
y
y
1
1
yy
yy
or factor 2 1
( 1)( 1)
y
y y
Multiply numerator and denominator by LCD
Ex 3: LCD = 2
11
11
x
x
2x
2
11
11
x
x
2
1
x
2
1
xDistribute
22
22
2
xx
xx
xx
2
2 1
x x
x
Factor ( 1)
( 1)( 1)
x x
x x
( 1)
x
x
Multiply numerator and denominator by LCD
Ex 4: Rewrite with positiveexponents
3
2 1
y y
y
Distribute
3
1
y
3
2
1
11
yy
y
LCD =
3y
3
2
1
11
yy
y
3
1
y 33
3
33
2
( )y
y yy
yy
y
4
31 y
y y
or factor
4
21
(1 )
y
y y
Multiply numerator and denominator by LCD
Ex 5: LCD =
2 5
1 4a b
a b
ab
1
ab
Distribute
2 5
1 4a b
a b
1
ab
2 5
4
ab aba bab aba b
2 5
4
b a
b a
Multiply numerator and denominator by LCD
Method 2: Simplify the numerator and denominator separately, then use division and multiply by reciprocal.
LCD =
Ex 6: 2 5
1 4a b
a b
ab
Add fractions in
numerator and
denominator.
2 5
1 4
b ab ab
a
a
ab
b
ba
2 5
4
b aabb aab
Multiply by reciprocal
2 5
4
b a ab
ab b a
2 5
4
b a
b a
Ex 7: 2
2
2
2
12
2 152 8
5 14
x x
x xx x
x x
( 4)( 3)( 5)( 3)( 4)( 2)( 7)( 2)
x xx xx xx x
Factor
( 4)( 5)( 4)( 7)
xxxx
Multiply by reciprocal
( 4) ( 7)
( 5) ( 4)
x x
x x
7
5
x
x
Simplify the numerator and denominator
Ex 8: 2
2
2
2
3 10
5 625
4 5
x x
x xx
x x
( 5)( 2)( 3)( 2)( 5)( 5)( 1)( 5)
x xx xx xx x
Factor
( 5)( 3)( 5)( 1)
xxxx
Multiply by reciprocal
( 5) ( 1)
( 3) ( 5)
x x
x x
1
3
x
x
Simplify the numerator and denominator
Ex 9: 3 3
2 2
2 2
2 22
a b
a ba ab b
a ab b
2 2
2 2
( )( )( )( )
( )( )
a b a ab ba b a b
a ab ba b a b
Factor
Multiply by reciprocal
2 2
2 2( ) ( )( )
( ) ( )
a ab b a b a b
a b a ab b
a b
2 2
2 2
( )( )
( )( )
a ab ba b
a ab ba b a b
Simplify the numerator and denominator
6-3 Summary
Method 1: Find LCD of all denominators, thenmultiply both the numerator and denominator by theLCD. 1
7
15
y
y
Method 2: Simplify the numerator and denominator separately, then use division and multiply by reciprocal.
2
2
2
2
12
2 152 8
5 14
x x
x xx x
x x
1
y1
1
yy
yy
1
y 1 7
1 5
y
y
( 4)( 3)( 5)( 3)( 4)( 2)( 7)( 2)
x xx xx xx x
( 4) ( 7)
( 5) ( 4)
x x
x x
( 4)( 5)( 4)( 7)
xxxx
7
5
x
x