# How do you write a rational expression with excluded values of -2 and 2?

May 2, 2017

$\frac{\text{insert any polynomial here}}{{x}^{2} - 4}$

#### Explanation:

This is kinda long but I hope you take the time to read the explanation.

A rational expression is simply a fraction where the numerator and denominator are polynomials.

In a rational expression you get excluded values when the denominator becomes equal to $0$ (because division by $0$ is undefined).

The numerator can be anything. It could be $1$, $10000 x + {x}^{2}$, whatever comes to your head. Any polynomial in the numerator will not give you an excluded value.

For these types of questions, it is actually very simple to answer problems like this. All you have to do is plug in the values you want to exclude into the expression $\left(x - \text{value}\right)$, then multiply all of these together.

Example

What is a function with excluded values $1 , - 2$?

For the numerator, we could use anything, for example ${x}^{2}$.

For the denominator, we plug in the excluded values into $\left(x - \text{value}\right)$. So since we have $1$ and $- 2$, we get $\left(x - 1\right)$ and $\left(x - \left(- 2\right)\right)$. Then we multiply these two together. $\left(x - 1\right) \left(x + 2\right) = {x}^{2} + x - 2$.

So we have:

${x}^{2} / \left({x}^{2} + x - 2\right)$

Now to answer your question, we just do the same. We want to exclude $- 2$ and $2$, so we get $\left(x - \left(- 2\right)\right)$ and $\left(x - 2\right)$. We multiply them together: $\left(x + 2\right) \left(x - 2\right) = {x}^{2} - 4$.

$\textcolor{red}{\frac{\text{insert any polynomial here}}{{x}^{2} - 4}}$