# Undefined Values

Whenever you're working with fractions, which rational expressions are, you always have to be concerend about the denominator. Remember that you can't divide anything by zero so finding the values that make a rational expression's denominator zero is an important step in any operation.

The basic procedure goes like this:

- Set the denominator equal to zero.
- Factor the denominator.
- Set the individual factors equal to zero and solve those equations.
- The solutions to the equations in step three are the values where the original reational expression is undefined.

If you need a refresher on how to solve polynomial equations, take a look at our short course page on the subject.

# Example 1

**For what values of x is $\frac{x^2+x+1}{x^2+3x+2}$ undefined?**

The expression is going to be undefined anywhere that the denominator, $x^2+3x+2$, equals zero, i.e. we need to solve the equation

$$x^2+3x+2=0$$

The first step is to factor the polynomial:

$$(x+2)(x+1)=0$$

The first equation gives us *x* = -2 and the second gives us *x* = -1 so our conclusion would be that the rational expression is undefined for both of those values.

# Example 2

**For what values of x is $\frac{x^2+x+1}{2x^2-x-3}$ undefined?**

The procedure here is the same as in Example 1. First, we need to factor the denominator:

$$(2x-3)(x+1)=0$$

The first equation gives us *x* = 3/2 and the second gives us *x* = -1 so our conclusion would be that the rational expression is undefined at those values.

# Example 3

**For what values of x is $\frac{x^2+2x+1}{2x^2+3}$ undefined?**

This is a tricky one. If you look carefully at the denominator, $2x^2+3$, you'll see that it can't be factored. When this happens, there are no values that make the denominator zero so there are no values for which the expression is undefined or, to phrase it positively, the expression is defined for all possible values of *x*.

**Directions:** This solution has 3 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.