Simplifying Rational Expressions
Rational Expressions
A Rational Expression is an algebraic fraction: a
fraction that contains a variable(s).
Our goal is to simplifying rational expressions by
“canceling” off common factors between the
numerator and denominator. Similar to simplifying a
numeric fraction.
Example:
2 x  13x  20 1
1


2
2 x  17 x  30 x  4 x  6
2
Simplifying Rational Expressions
Simplify the following expressions by finding a
common factor:
16 x
16 x
1
x 3
x 3
x
x
x 5
x 1

1
2
x
2
x
 1 
x
3
x
3
x 2
x 2

x 5
x 1
x
3
1 
x 5
x 1
1
The Major Requirement for Simplifying Rational
Expressions
A fellow student simplifies the following
expressions:
 4 1  4
4 x
4x

4

1

5
x
x
Which simplification is correct? Substitute two
values of x into each to justify your answer.
4 3
3
Equal.

12
3
4
Not Equal.
43
3
  2.3
7
3
MUST BE MUITLIPLICATION! It can be simplified if the numerator and denominator
are single terms and are product of factors.
Which is Simplified Correctly?
Which of the following expressions is simplified correctly?
Explain how you know.
x 2  x 3
x 3
x
X
Left
Right
-5
-11.5
25
-1
1.5
1
0
1
0
4
3.29
16
7
5.9
49
2
 x  2 x 3
x 3
The left side
of the
equation has
to equal the
right.
 x2
X
Left
Right
-5
-3
-3
-1
1
1
0
2
2
4
6
6
7
9
9
MUST BE MUITLIPLICATION! It can be simplified if the numerator and denominator
are single terms and are product of factors.
Example 1
State the values that make
the denominator zero and
then simplify:
Make the Denominator
0:
2 and -7.
These Make the
ORIGINAL denominator
equal 0. We assume that
x can never be these
values.
3  x  2  x  7 
15  x  2  x  7 
2
Half the work is done. It is
already factored.
3  x  2  x  7  x  7 
15  x  2  x  7 
3
3 x  7
15 3
x7
5
Rewrite
CAN cancel since the top
and bottom have
common factors.
Don’t forget about
numeric Factors.
Example 2
State the values that make
the denominator zero and
then simplify:
Make the Denominator
0:
4, -4, and 0.
These Make the
ORIGINAL denominator
equal 0. We assume that
x can never be these
values.
2 x 2  3 x  20
4 x3  64 x
Can NOT cancel since its not in
factored form
4 x  x 2  16 
Always Factor
Completely
 2 x  5 x  4 
 2 x  5 x  4 
4 x  x  4  x  4 
CAN cancel since the top
and bottom have a
common factor
2x  5
2x  5
or
4x  x  4
4 x 2  16 x
Example 3
State the values that make
the denominator zero and
then simplify:
Make the Denominator
0:
a=0 or b=0
These Make the
ORIGINAL denominator
equal 0. We assume that
a & b can never be these
values.
64ab3  24a 2b2
16a 4b5
8ab 2 8b 3a
2
8ab 2a 3b3 
8b  3a
3 3
2a b
Can NOT cancel since its not in
factored form
If they are not
quadratics, find a
common factor.
CAN cancel since the top
and bottom have a
common factor