Rational Expressions

Section 0.5

Rational Expressions and domain restrictions Rational number- ratio of two integers with

the denominator not equal to zero. Rational expression- ratio or quotient of two

polynomials with the denominator not equal to zero

Examples: Rational number:

Rational expression:

where x = 6

2

152 7

6

x

x

Domain- set of real numbers that your algebraic expression is defined. Think about domain as what values

are OK to plug into your equation. For rational expressions our domain

will not be defined for the values that make the denominator zero.

What is the domain for:

Answer: All Real numbers except x = -3

2 8

3

x

x

Find the domain for each algebraic expression

Domain: All real numbers

Domain: All real numbers except x = 0

Domain: All real numbers except

Domain: All real numbers

x

x

8

3

x

x

3

5 6

2 45x

x 6

5

6

42

x

x

Find the domain for each algebraic expression

Domain: All real numbers except x = 0 and x = 5

Domain: All real numbers except x = 4 and x = -4

2 8

52

x

x x

6 5

162

x

x

2 8

5

x

x x( )

6 5

4 4

x

x x( )( )

Reduce the rational expression

x x

x

2 8 15

3 15

( )( )

( )

x x

x

5 3

3 5

( )x 3

3

Where x = -1

5 3 2

2 4 2

2

2

x x

x x

( )( )

( )( )

5 2 1

2 1 1

x x

x x

( )

( )

5 2

2 1

x

x

Where x = -5

Multiply the rational expressions and simplify

4

4

16

8 2

2 2

2

x

x

x

x

2

2

42

44

4

4

x

xx

x

x

2

4x

Check domain at factored step:

04 x

4x

042 x

42 x

4x

4xDomain: All reals except:

Multiply

39

23

44

13 23

2

x

xxx

xx

x

133

23

14

13 2

x

xxx

xx

x

133

21

14

13

x

xxx

xx

x

12

2x

Domain Restrictions:

04 x 01x 013 x

Divide the rational expressions

3

32

5

1234

x

xx

x

x

xx

x

x

x

123

543

32

223

522 3

xxx

x

x

xx

3

5xDomain: All reals except -2, 0, and 2

Divide

4

32

7

1829

x

xx

x

x

92

792

42

xx

x

x

x

332

733 4

xxx

x

x

xx

2

7 2x Domain: All reals except 0, 3 and -3