Revised 11/09
1
HFCC Math Lab
Intermediate Algebra
–
9
SIMPLIFYING COMPLEX RATIONAL EXPRESSIONS
A complex rational expression is a rational expression in which either the numerator or the
denominator or both the numerator and the denominator involve rational expressions.
For
example:
1)
2
1
4
1
x
x
x
is a complex rational expression because its numerator
1
4
x
x
is
itself a rational expression.
2)
5
2
10
x
x
x
is a complex rational expression because its denominator
2
10
x
x
is itself a rational expression.
3)
2
2
2
6
2
8
4
12
x
x
x
x
x
is a complex rational expression.
Its numerator is
2
2
6
x
x
and
its denominator is
2
2
8
4
12
x
x
x
.
4)
1
2
2
2
y
x
x
y
y
x
is a complex rational expression.
Its numerator is
1
2
y
x
and
its denominator is
2
2
x
y
y
x
.

Revised 11/09
2
NOTE
:
It should be quite clear from the above four examples that every complex rational
expression involves at least two fraction lines.
The longest fraction line in this
handout and most mathematics teachers lecture notes on the blackboard separates
the numerator from the denominator.
What you see above this longest fraction
line,
is the numerator and what you see below this longest fraction line,
is the
denominator.
In most text books,
the longest fraction line is replaced by the
darkest fraction line.
The following two methods will be used in this handout to
simplify complex rational expressions.
Method I
:
If both the numerator and denominator of the complex rational expression contain
only one rational expression, we divide the numerator by the denominator
and simplify.