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Undefined Rational Expressions

Determining When a Rational Expression is Undefined

A rational number is a number that can be written in the form where a and b are integers and b 0.

For example, are rational numbers.

We define a rational expression in a similar manner.

 

Definition — Rational Expression

A rational expression is an expression that can be written in the form where P and Q are polynomials and Q 0.

 

Note:

The integer -2 is a rational number, since it can be written as Likewise, 1.5 is a rational number, since it can be written as

Remember, constants such as 2 and 7 are monomials of degree 0. So, is an example of a rational expression.

 

Here are some examples of rational expressions:

The denominator of a rational expression cannot equal 0. This is because division by 0 is undefined. Therefore, it is important to determine the values of the variable that make the denominator 0. We say that the rational expression is undefined for those values.

 

Example 1

Find the value(s) of x for which this rational expression is undefined:

Solution

The rational expression is undefined when its denominator is 0.

Set the denominator equal to 0.

Then solve for x.

Subtract 8 from both sides.

x + 8 = 0

 

x = -8

When x = -8, the denominator is 0.
We have
Therefore, is undefined when x = -8.

 

Note:

 It’s okay for the numerator of a rational expression to equal 0. For example, is defined when x = 0.

However, is not defined when x = 5.

undefined

 

Example 2

Find the value(s) of x for which this rational expression is undefined:

Solution

Set the denominator equal to zero.

Then solve for x.

Factor the left side of the equation.

Use the Zero Product Property.

Solve these equations.

 x2 - 9

(x - 3)(x + 3)

x - 3 = 0

x =3 

= 0

= 0

or

or

 

 

x + 3 = 0

x = -3

Therefore, is undefined when x = 3 or x = -3.
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