Simplifying Rational
Expressions
Simplifying a Fraction
Simplify:
25
175
7

75 25 3
The techniques we use to simplify a fraction without variables
(Finding the greatest common FACTOR) is the same we will use
to simplify fractions with variables.
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
We will see how to simplify the original expression.
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
1
x
3
1 
x 5
x 1
Can we simplify this one more?
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
Equal.
Not Equal.
answer.
4 3
3

12
3
4
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?
Make a table for each side of the
equation to see if they are the
same.
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.
Back to a Previous Example…
Can we simplify the following expression
more?
x 5
x 1

x 2
x 2

x 5
x 1
1 
x 5
x 1
NO!! IT MUST BE MUITLIPLICATION! The numerator is the sum of two
terms. The denominator is the difference of two terms. The numerator and
the denominator are not written as a product. Also, there greatest
common factor is 1.
If you still disagree, make a table to check your hypothesis.
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 any factors if
they are raised to a
power
CAN cancel since the
top and bottom have
common factors.
Don’t forget about
cancelling common
numeric Factors.
Example 2
State the values that make
the denominator zero and
then simplify:
The denominator is factored, so it is
obvious what values of x make it 0
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  3 x  20
4 x 3  64 x
2
 2 x  5 x  4 
4 x  x 2  16 
 2x  5 x  4
4 x  x  4  x  4 
Can NOT cancel since its not
in factored form. Also it is not
obvious what values of x make
the denominator 0.
Always Factor
Completely
CAN cancel since the
top and bottom have a
common factor
2x  5
2x  5
or
4x  x  4
4 x 2  16 x
Finding where the
fraction undefined is the
same as finding when
the denominator is 0.
State the values that make
the fraction undefined and
then simplify:
Make the
Fraction Undefined:
a=0 or b=0
These Make the
ORIGINAL
denominator equal 0.
We assume that a & b
can never be these
values.
Example 3
64ab3  24a 2b2
16a 4b5
8ab 2 8b 3a
2
8ab 2a 3b3 
8b  3a
3 3
2a b
Can NOT cancel any
factors since its not in
factored form
If they are not
quadratics, find a
common factor.
CAN cancel common
factors since the top
and bottom have a
common factor