lecture complex fractions

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 6.3

ComplexRational

Expressions

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 2

Objective #1 Simplify complex rational expressions by

multiplying by 1.

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.

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

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.

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.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7

Simplifying Complex Fractions

Simplify.

CONTINUED

1

5x

Simplify.5x

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

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

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

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

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

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

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

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15

Objective #2 Simplify complex rational expressions by dividing.

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.

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

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

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

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

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

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

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

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

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

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

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