This document provides information on simplifying rational expressions. It defines rational expressions as fractions with polynomial numerators and denominators. The key steps to simplify rational expressions are to divide out common factors between the numerator and denominator, factor any polynomials, and then divide the common factors. Examples are provided to demonstrate these steps. Restrictions on rational expressions refer to values that make the denominator equal to zero. Students are given practice problems to simplify rational expressions and identify restrictions.

1. Rational Expressions
Simplifying
Joyce DuVall
Green Valley High School
Henderson, Nevada

2. Simplifying Rational Expressions
 The objective is to be able to simplify a
rational expression
5
2x 
3
92
x
x 

3.  Polynomial – The sum or difference of monomials.
 Rational expression – A fraction whose numerator
and denominator are polynomials.
 Domain of a rational expression – the set of all
real numbers except those for which the denominator
is zero.
 Reduced form – a rational expression in which the
numerator and denominator have no factors in
common.

4.  Divide out the
common factors
 Factor the
numerator and
denominator and
then divide the
common factors

5. Dividing Out Common Factors
Step 1 – Identify any factors which are common to both the
numerator and the denominator.
5
5 7
x
x( )
The numerator and denominator
have a common factor.
The common factor is the five.

6. Dividing Out Common Factors
Step 2 – Divide out the common factors.
The fives can be divided since 5/5 = 1
The x remains in the numerator.
The (x-7) remains in the denominator
5
5 7
x
x( )

x
x  7

7. Factoring the Numerator
and Denominator
Factor the numerator.
Factor the denominator.
Divide out the common factors.
Write in simplified form.
3 9
12
2
3
x x
x


8. Factoring
Step 1: Look for common factors
to both terms in the numerator.
3 9
12
2
3
x x
x
 3 is a factor of both 3 and 9.
X is a factor of both x2 and x.
Step 2: Factor the numerator.
3 9
12
2
3
x x
x
 3 3
12 3
x x
x
( )

9. Factoring
Step 3: Look for common factors to the
terms in the denominator and factor.
3 9
12
2
3
x x
x

The denominator only has one term.
The 12 and x3 can be factored.
The 12 can be factored into 3 x 4.
The x3 can be written as x • x2.
3 9
12
2
3
x x
x
 3 3
3 4 2
x x
x x
( )
  

10. Divide and Simplify
Step 4: Divide out the common factors. In this
case, the common factors divide to become 1.
3 3
3 4 2
x x
x x
( )
  
Step 5: Write in simplified form.
x
x
 3
4 2

11. You Try It
Simplify the following rational expressions.
1
9
24
2
2
.
x yz
xyz
2
3
4 32
.
a
a a

 
3
3 15
7 102
.
x
x x

 
4
2 15
12
2
2
.
x x
x x
 
 
5
14 35 21
12 30 18
2
2
.
x x
x x
 
 

12. Restrictions on
Rational Expressions
For what value of x is undefined?x x
x
2
2 15
4 20
 

It is undefined for any value of “x” which makes the
denominator zero.
4 20 0x  
4 5 0( )x  
x  5 0
x    5 5 0 5
x  5
The restriction is
that x cannot equal
5.

13. YOU TRY IT
What are the excluded values of the variables for
the following rational expressions?
1
42
14
3 2
2 3
.
y z
y z
2
36
2 12
2
.
x
x


3
4 12
2 8
2
2
.
c c
c c
 
 