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
2
Examples: Rational number: 15
Rational expression: 2 x  7
x6
where x = 6

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: 2 x  8

x3
Answer: All Real numbers except x = -3
Find the domain for each
algebraic expression
2 x  4 Domain: All real numbers
5
x 8
3x
Domain: All real numbers except x = 0
x3
5x  6
Domain: All real numbers except
6x
Domain: All real numbers
2
x 4
6
x
5
Find the domain for each
algebraic expression
2x  8
x 2  5x
2x  8

x ( x  5)
Domain: All real numbers except x = 0
and x = 5
6x  5
2
x  16
6x  5

( x  4)( x  4)
Domain: All real numbers except x = 4
and x = -4
Reduce the rational
expression
x  8 x  15 ( x  5)( x  3)
( x  3)


3( x  5)
3
3x  15
2
Where x = -5
5 x 2  3x  2
2x2  4x  2
(5x  2)( x  1)

2( x  1)( x  1)
(5x  2)

2( x  1)
Where x = -1
Multiply the rational
expressions and simplify
4  x x  16

2
4  x 8  2x
2
2
Check domain at
factored step:
4 x  0
4  x x  4x  4

4 x
2 4  x2
2

x4
2
x  4

x2  4  0
x 2  4
x   4
Domain: All reals
except:
x  4
x  2
Multiply
12
3x  1 x 3  3x 2  2 x

2
4x  4x
9x  3

3x  1 x x  3x  2

4 xx  1
33x  1
2

3x  1 xx  1x  2

4 xif!x  1
33x  1
Domain Restrictions:
4x  0
x 1  0
3x  1  0
Divide the rational
expressions
x 2  4 3x3  12x

x
5 x3
x2  4
5 x3
 3
x
3x  12x
x  2x  2 
x
5x
3
3
5x
3xx  2x  2
Domain: All reals except -2, 0, and 2
Divide
x 2  9 2 x3  18x

x
7 x4
x2  9
7 x4

x
2x x2  9

x  3x  3 
x

7 x4
2 xx  3x  3
7 x2
2
Domain: All reals except 0, 3 and -3
Adding Rational Expressions
1 5

7 7
6
7
Need to reduce
x7
3x  1

2
x  2 x  22
x  7   3x  1
x  22
4x  8
x  22
4x  2
x  22
4
x2
D: all reals except x = -2
Adding Rational Expressions
3 5

4 7
Need a Common
Denominator
21 20

28 28
41
28
3 x
x

2x 1 x 1
3  x x  1  x2 x  1
2 x  1x  1 x  12 x  1
3  x x  1  x2 x  1
2 x  1x  1
3x  3  x 2  x  2 x 2  x
2 x  1x  1
x 2  5x  3
2 x  1x  1
D: All reals except
x = -1/2 and x = 1