Pre-Calculus 11
Rational Expressions
p
, where p and q are polynomials and q  0 .
q
x
x 4
2
2
x  5x
x  3x  28
A rational expression is an algebraic fraction of the form
Examples :
5
a
6r 2 p 3
4rp 4
Whenever we are working with rational expressions, we must identify the non-permissible values of the
variables. A non-permissible value is a value of the variable that causes the expression to be undefined. For
rational expressions, this occurs when the denominator is equal to 0.
When simplifying rational expressions, note that it may be convenient to multiply or divide by an equivalent
form of “1”, which will not change the value of the original expression. For example, consider the expression:
5x
(Note that x  2 )
x 2
You can multiply this expression by, say,
5x
x
5x 2

 2
(x  2) x
x  2x
x
1
or
, which is actually the same as multiplying by 1.
1
x
5x
1
5x
5x



(x  2) 1
x  2 2  x
OR
5x
5x
5x 2
Thus,
and 2
and
are equivalent rational expressions.
x 2
2x
x  2x
Example 1: Determine Non-Permissible Values
For each rational expression, determine all non-permissible values.
5z
a.
7xy 2
4x
b.
5x 2  4x
x 2  9x  14
c.
x 2  25
Solution:
Note that denominators cannot equal zero.
a.
5z
7xy 2
b.
4x
5x 2  4x
c.
x 2  9x  14
x 2  25
Pre-Calculus 11
Example 2: Simplify a Rational Expression
To simplify a rational expression, factor the numerator and denominator. Then divide both the numerator and
denominator by any factors that are common to both. When a rational expression is in simplest form, the
numerator and denominator will have no common factors other than 1. Consider the factors of the
denominator to find the non-permissible values before simplifying the expression.
Simplify each rational expression. State the non-permissible values.
4x 2y 3
a.
20x 3y
Solution:
a.
4x 2y 3
20x 3y
x 2  3x  18
b. 2
x  10x  24
c.
4x 2  1
2x 2  5x  3
x 2  3x  18
b. 2
x  10x  24
4x 2  1
c.
2x 2  5x  3
Pre-Calculus 11
Example 3: Recognize Additive Inverses
Simplify each rational expression and state the non-permissible values.
a.
9x
x 2  81
b.
10  2x
x  x  20
2
Solution:
a.
9x
x 2  81

(9  x )
(x  9)(x  9)
Note that the numerator is the additive inverse of one of the factors in the denominator.
Create an equivalent rational expression by multiplying by

(9  x )
1
9  x


(x  9)(x  9) 1
1(x  9)( x  9)

x 9
1
1


, x  9
1(x  9)(x  9)
1(x  9)
x 9
b.
10  2x
x  x  20
2
1
.
1
Pre-Calculus 11
Example 4: Rational Expressions with Pairs of Non-Permissible Values
4x 2y  10xy
Consider the expression
.
6xy 2  8x 2y
a. What are the non-permissible values of x? What are the non-permissible values of y?
b. Simplify the rational expression.
c. Evaluate the expression for x = 2.6 and y = 1.2.
Solution:
a. What are the non-permissible values of x? What are the non-permissible values of y?
Examples of non-permissible values are ( 0 , _____ ) , ( _____ , 1 ) , ( 2, _____ )
b. Simplify the rational expression.
4x 2y  10xy
6xy 2  8x 2y

c. Evaluate the expression for x = 2.6 and y = 1.2.