# Intro to rationalÂ expressions

CCSS Math: HSA.APR.D.7, HSF.IF.B.5

Learn what rational expressions are and about the values for which they are undefined.

#### What you will learn in this lesson

This lesson will introduce you to rational expressions. You will learn how to determine when a rational expression is undefined and how to find its domain.

## What is a rational expression?

A

**polynomial**is an expression that consists of a sum of terms containing integer powers of $x$, like $3x^2-6x-1$.A

**rational expression**is simply a*quotient*of two polynomials. Or in other words, it is a fraction whose numerator and denominator are polynomials.These are examples of rational expressions:

$\dfrac{1}{x}$, $\quad\dfrac{x+5}{x^2-4x+4}$, $\quad\dfrac{x(x+1)(2x-3)}{x-6}$

Notice that the numerator can be a constant and that the polynomials can be of varying degrees and in multiple forms.

## Rational expressions and undefined values

Consider the rational expression $\dfrac{2x+3}{x-2}$.

We can determine the value of this expression for particular $x$-values. For example, let's evaluate the expression at $\blueD{x}=\blueD1$.

From this, we see that the value of the expression at $\blueD{x}=\blueD{1}$ is $\goldD{-5}$.

Now let's find the value of the expression at $\blueD{x}=\blueD{2}$.

An input of $2$ makes the denominator $0$. Since division by $0$ is undefined, $\blueD x=\blueD 2$ is not a possible input for this expression!

## Domain of rational expressions

The

**domain**of any expression is the set of all possible input values.In the case of rational expressions, we can input any value except for those that make the denominator equal to $0$ (since division by $0$ is undefined).

In other words, the

**domain of a rational expression**includes all real numbers except for those that make its denominator zero.### Example: Finding the domain of $\dfrac{x+1}{(x-3)(x+4)}$

Let's find the zeros of the denominator and then restrict these values:

So we write that the domain is

*all real numbers except $\textit 3$ and $\textit{-4}$*, or simply $x\neq 3, -4$.