lecture complex fractions
TRANSCRIPT
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Section 6.3
ComplexRational
Expressions
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Objective #1 Simplify complex rational expressions by
multiplying by 1.
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Simplifying Complex Fractions
Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more fractions.
551 15
xx
x
Woe is me,
for I am complex.
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Complex Rational Expressions
Simplifying a Complex Rational Expression by Multiplying by 1 in the
Form T
1) Find the LCD of all rational expressions within the complex rational expression.
2) Multiply both the numerator and the denominator of the complex rational expression by this LCD.
3) Use the distributive property and multiply each term in the numerator and denominator by this LCD. Simplify each term. No fractional expressions should remain within the numerator and denominator of the main fraction.
4) If possible, factor and simplify.
LCD
LCD
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Simplifying Complex Fractions
EXAMPLE
Simplify: .1
51
55
x
xx
SOLUTION
The denominators in the complex rational expression are 5 and x. The LCD is 5x. Multiply both the numerator and the denominator of the complex rational expression by 5x.
x
xx
x
x
x
xx
151
55
5
51
51
55 Multiply the numerator and
denominator by 5x.
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Simplifying Complex Fractions
xxx
xx
xx
15
5
15
55
55
Use the distributive
property.
CONTINUED
xxx
xx
xx
15
5
15
55
55
Divide out common factors.
5
252
x
xSimplify.
51
55
x
xxFactor and simplify.
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Simplifying Complex Fractions
Simplify.
CONTINUED
1
5x
Simplify.5x
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Simplifying Complex Fractions
EXAMPLE
Simplify: .6
16
1xx
SOLUTION
The denominators in the complex rational expression are x + 6 and x. The LCD is (x + 6)x. Multiply both the numerator and the denominator of the complex rational expression by (x + 6)x.
Multiply the numerator and denominator by (x + 6)x.
6
1
6
1
6
6
6
1
6
1
xxxx
xxxx
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Simplifying Complex Fractions
Use the distributive property.
66
16
6
16
xxx
xxx
xx
CONTINUED
Divide out common factors.
66
16
6
16
xxx
xxx
xx
Simplify.
66
6
xx
xx
Simplify.xx
xx
366
62
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Simplifying Complex Fractions
CONTINUED
Subtract.xx 366
62
Factor and simplify.
66
16
xx
Simplify. 6
1
xx
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1a. Simplify: 2
2
1
1
x
y
x
y
Multiply the numerator and denominator by the LCD of 2.y
22
2
2 2 2 2 22
2 2 2
1 1 1
1 1 1
x x x yy
yy y y
x y x x yy
y y y
2
2 2
( )
( )( )
xy y y x y y
x y x y x yx y
Objective #1: Example
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1a. Simplify: 2
2
1
1
x
y
x
y
Multiply the numerator and denominator by the LCD of 2.y
22
2
2 2 2 2 22
2 2 2
1 1 1
1 1 1
x x x yy
yy y y
x y x x yy
y y y
2
2 2
( )
( )( )
xy y y x y y
x y x y x yx y
Objective #1: Example
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Objective #1: Example
1b. Simplify:
1 177
x x
Multiply the numerator and denominator by the LCD of ( 7).x x
1 1 1 1 ( 7) ( 7)( 7)7 7 7
7 ( 7) 7 7 ( 7)
( 7) 7 77 ( 7) 7 ( 7) 7 ( 7)
1 1( 7) ( 7)
x x x xx xx x x x x xx x x x
x x x xx x x x x x
x x x x
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Objective #1: Example
1b. Simplify:
1 177
x x
Multiply the numerator and denominator by the LCD of ( 7).x x
1 1 1 1 ( 7) ( 7)( 7)7 7 7
7 ( 7) 7 7 ( 7)
( 7) 7 77 ( 7) 7 ( 7) 7 ( 7)
1 1( 7) ( 7)
x x x xx xx x x x x xx x x x
x x x xx x x x x x
x x x x
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Objective #2 Simplify complex rational expressions by dividing.
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Simplifying Complex Fractions
Simplifying a Complex Rational Expression by Dividing
1) If necessary, add or subtract to get a single rational expression in the numerator.
2) If necessary, add or subtract to get a single rational expression in the denominator.
3) Perform the division indicated by the main fraction bar: Invert the denominator of the complex rational expression and multiply.
4) If possible, simplify.
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Simplifying Complex Fractions
EXAMPLE
Simplify: .
6653
442
9
22
22
mmm
mm
mmmm
SOLUTION
1) Subtract to get a single rational expression in the numerator.
222 2
2
3344
2
9
mmm
m
mmm
m
2
2
22
2
233
3322
332
332
233
2
mmm
mmmm
mmm
mm
mmm
mm
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Simplifying Complex Fractions
2) Add to get a single rational expression in the denominator.
2
23
2
223
233
1846
233
18244
mmm
mmm
mmm
mmmm
3223
3
665
322
mm
m
mmmm
m
mm
323
333
332
3
323
33
mmm
mmm
mmm
mm
mmm
m
323
9
323
393 22
mmm
m
mmm
mmm
CONTINUED
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Simplifying Complex Fractions
3) & 4) Perform the division indicated by the main fraction bar: Invert and multiply. If possible, simplify.
3239
233
1846
6653
442
92
2
23
22
22
mmmm
mmm
mmm
mmm
mm
mmmm
9
323
233
184622
23
m
mmm
mmm
mmm
9
323
233
184622
23
m
mmm
mmm
mmm
92
18462
23
mm
mmm
CONTINUED
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2a. Simplify:
1 11 11 11 1
x xx xx xx x
The LCD of the numerator is ( 1)( 1)x x .
The LCD of the denominator is ( 1)( 1)x x .
Objective #2: Example
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2a. Simplify:
1 11 11 11 1
x xx xx xx x
The LCD of the numerator is ( 1)( 1)x x .
The LCD of the denominator is ( 1)( 1)x x .
Objective #2: Example
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Objective #2: Example
2 2
2 2
2 2
( 1)( 1) ( 1)( 1) 2 1 2 11 1( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1)1 1
1 1 ( 1)( 1) ( 1)( 1) 2 1 2 11 1 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1)
2 1 ( 2 1)(
x x x x x x x xx xx x x x x x x xx x
x x x x x x x x x xx x x x x x x x x x
x x x xx
2 2
2 2 2 2
2 1 2 11)( 1) ( 1)( 1)
2 1 2 1 2 1 2 1( 1)( 1) ( 1)( 1)
4( 1)( 1) ( 1)( 1)4 4
2 2 2( 1)( 1)2 2 2 2 2( 1)( 1)( 1)
22 1
x x x xx x x
x x x x x x x xx x x xx
x x x xx xx xx x x
x xx
x
CONTINUED
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Objective #2: Example
2 2
2 2
2 2
( 1)( 1) ( 1)( 1) 2 1 2 11 1( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1)1 1
1 1 ( 1)( 1) ( 1)( 1) 2 1 2 11 1 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1)
2 1 ( 2 1)(
x x x x x x x xx xx x x x x x x xx x
x x x x x x x x x xx x x x x x x x x x
x x x xx
2 2
2 2 2 2
2 1 2 11)( 1) ( 1)( 1)
2 1 2 1 2 1 2 1( 1)( 1) ( 1)( 1)
4( 1)( 1) ( 1)( 1)4 4
2 2 2( 1)( 1)2 2 2 2 2( 1)( 1)( 1)
22 1
x x x xx x x
x x x x x x x xx x x xx
x x x xx xx xx x x
x xx
x
CONTINUED
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Objective #2: Example
2b. Simplify: 2
1 21 4
1 7 10
x
x x
Rewrite the expression without negative exponents.
Then multiply the numerator and denominator
by the LCD of 2.x
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Objective #2: Example
2b. Simplify: 2
1 21 4
1 7 10
x
x x
Rewrite the expression without negative exponents.
Then multiply the numerator and denominator
by the LCD of 2.x
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Objective #2: Example
2 2
1 2
2
22
2 2 2
2 2 22
2 2
2
2
41
1 47 101 7 10 1
4 41 1
7 10 7 101 1
4 ( 2)( 2) 2( 5)( 2) 57 10
x xx x
x x
xx
x x xx x x
xx x x x
x x x xx x xx x
CONTINUED
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Objective #2: Example
2 2
1 2
2
22
2 2 2
2 2 22
2 2
2
2
41
1 47 101 7 10 1
4 41 1
7 10 7 101 1
4 ( 2)( 2) 2( 5)( 2) 57 10
x xx x
x x
xx
x x xx x x
xx x x x
x x x xx x xx x
CONTINUED