Slide 7- 1Copyright © 2012 Pearson Education, Inc.

Copyright © 2012 Pearson Education, Inc.

6.3 Complex Rational

Expressions

■ Multiplying by the LCD

■ Using Division to Simplify.

Slide 6- 3Copyright © 2012 Pearson Education, Inc.

To Simplify a Complex Rational Expression by Multiplying by the LCD

1. Find the LCD of all rational expressions within the complex rational expression.

2. Multiply the complex rational expression by an expression equal to 1. Write 1 as the LCD over itself (LCD/LCD).

3. Simplify. No fraction expressions should remain within the complex rational expression.

4. Factor and, if possible, simplify.

Slide 6- 4Copyright © 2012 Pearson Education, Inc.

Example Simplify:

Solution Here we look for the LCD of all four factors.

1 3

.2 1a b

a b

1 3 1 3

2 1 2 1ab

aa b

ba b

a b a b

1 31 3

2 1 2 1

abab a ba b

a b a bab ab

Multiplying by a factor equal to 1, using the LCD: ab/ab = 1.

Multiplying the numerator by ab

Don’t forget the parentheses!

Multiplying the denominator by ab

Slide 6- 5Copyright © 2012 Pearson Education, Inc.

Solution continued

1 31 3

2 1 2 1

abab a ba b

a b a bab ab

1 3( ) ( )

2 1( ) ( )

ab ab

ab

a b

a bab

3

2

b a

b a

Using the distributive law

Simplifying

Slide 6- 6Copyright © 2012 Pearson Education, Inc.

To Simplify a Complex Rational Expression by Dividing

1. Add or subtract, as necessary, to get a single rational expression in the numerator.

2. Add or subtract, as necessary, to get a single rational expression in the denominator.

3. Divide the numerator by the denominator (invert and multiply).

4. If possible, simplify by removing a factor equal to 1.

Slide 6- 7Copyright © 2012 Pearson Education, Inc.

Example Simplify:

Solution

4 .3

2 8

xx

x

343 4 2 8

2 8

xxx

x xx

2 8

4 3

x x

x

4

x

x

2( 4x

)

3

2

3

x

Rewriting with a division symbol

Multiplying by the reciprocal of the divisor (inverting and multiplying)

Factoring and removing a factor equal to 1.

Slide 6- 8Copyright © 2012 Pearson Education, Inc.

Example Simplify:

Solution

53 .

4 12

x

x x

33

5 53 3

4 1 4 1222 2

x x

x x x x

8 1 72 2

15 13 3

2

53

x x

x x

x

Multiplying by 1 to get the LCD, 3, for the numerator.

Multiplying by 1 to get the LCD, 2x, for the denominator.

Adding

Subtracting

Slide 6- 9Copyright © 2012 Pearson Education, Inc.

Solution continued

15 153 3 3

8 1 72 2 2

x x

x x x

15 7

3 2

x

x

15 2

3 7

x x

230 2

21

x x

Rewriting with a division symbol. This is often done mentally.

Multiplying by the reciprocal of the divisor (inverting and multiplying).

Slide 6- 10Copyright © 2012 Pearson Education, Inc.

Example Simplify: Solution The LCD is x.

When we multiply by x, all fractions in the numerator and denominator of the complex rational expression are cleared:

11

11

x

x

1 11 1

1 11 1

x x

x

x

xx

11( ) ( )

11( ) ( )

x x

x

x

xx

Using the distributive law

11( ) ( ) 1

1 11( ) ( )

x x xxxx x

x

Slide 6- 11Copyright © 2012 Pearson Education, Inc.

Example Simplify: Solution

3 2

2

6 2

.3 4x x

x x

3 2 3 2

2 2

3

3

6 2 6 2

3 4 3 4x x x x

x

x

xx

x x

The LCD is x3 so we multiply by 1 using x3/x3.

3 33 2

32

3

6 2

3 4x x

x

x x

x xx

2

6 2 2(3 )

3 4 (3 4)

x x

x x x x

Using the distributive law

All fractions have been cleared and simplified.

Slide 6- 12Copyright © 2012 Pearson Education, Inc.

If negative exponents occur, we first find an equivalent expression using positive exponents and then proceed as in the preceding examples.

There is no one method that is best. Either method can be used with any complex rational expression.