Download - Lecture complex fractions
![Page 1: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/1.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
Section 6.3
ComplexRational
Expressions
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
![Page 2: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/2.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 2
Objective #1 Simplify complex rational expressions by
multiplying by 1.
![Page 3: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/3.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 3
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.
![Page 4: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/4.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4
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
![Page 5: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/5.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5
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.
![Page 6: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/6.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6
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.
![Page 7: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/7.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7
Simplifying Complex Fractions
Simplify.
CONTINUED
1
5x
Simplify.5x
![Page 8: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/8.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8
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
![Page 9: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/9.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9
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
![Page 10: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/10.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10
Simplifying Complex Fractions
CONTINUED
Subtract.xx 366
62
Factor and simplify.
66
16
xx
Simplify. 6
1
xx
![Page 11: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/11.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11
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
![Page 12: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/12.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12
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
![Page 13: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/13.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13
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
![Page 14: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/14.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14
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
![Page 15: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/15.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15
Objective #2 Simplify complex rational expressions by dividing.
![Page 16: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/16.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16
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.
![Page 17: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/17.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17
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
![Page 18: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/18.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18
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
![Page 19: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/19.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19
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
![Page 20: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/20.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 20
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
![Page 21: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/21.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 21
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
![Page 22: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/22.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 22
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
![Page 23: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/23.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 23
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
![Page 24: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/24.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24
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
![Page 25: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/25.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25
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
![Page 26: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/26.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 26
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
![Page 27: Lecture complex fractions](https://reader030.vdocuments.site/reader030/viewer/2022032513/55d3558bbb61ebd6148b45af/html5/thumbnails/27.jpg)
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 27
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