Intro to rational expressions (article)

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

3 x^{2} - 6 x - 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:

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

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

\frac{2 x + 3}{x - 2}

We can determine the value of this expression for particular

x

-values. For example, let's evaluate the expression at

x = 1

\begin{aligned} \frac{2 (1) + 3}{1 - 2} & = \frac{5}{- 1} \\ = - 5 \end{aligned}

From this, we see that the value of the expression at

x = 1

- 5

Now let's find the value of the expression at

x = 2

\begin{aligned} \frac{2 (2) + 3}{2 - 2} & = \frac{7}{0} \\ = undefined! \end{aligned}

An input of

2

makes the denominator

0

. Since division by

0

is undefined,

x = 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 $\frac{x + 1}{(x - 3) (x + 4)}$ ‍

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

\begin{aligned} (x - 3) & (x + 4) = 0 \\ ↙ & ↘ \\ x - 3 = 0 & or x + 4 = 0 \\ x = 3 & or x = - 4 \end{aligned}

So we write that the domain is all real numbers except $3$ ‍ and $-4$ ‍, or simply

x \neq 3, - 4

Check your understanding

Problem 1

What is the domain of $\frac{x + 1}{x - 7}$ ‍?

Problem 2

What is the domain of $\frac{3 x - 7}{2 x + 1}$ ‍?

Problem 3

What is the domain of $\frac{2 x - 3}{x (x + 1)}$ ‍?

Problem 4

What is the domain of $\frac{x - 3}{x^{2} - 2 x - 8}$ ‍?

Problem 5

What is the domain of $\frac{x + 2}{x^{2} + 4}$ ‍?

Course: Algebra (all content) > Unit 13

Intro to rational expressions

What you will learn in this lesson

What is a rational expression?

Rational expressions and undefined values

Domain of rational expressions

Example: Finding the domain of $\frac{x + 1}{(x - 3) (x + 4)}$ ‍

Check your understanding

Want to join the conversation?

Algebra (all content)

Course: Algebra (all content) > Unit 13

What you will learn in this lesson

What is a rational expression?

Rational expressions and undefined values

Domain of rational expressions

Example: Finding the domain of x+1(x−3)(x+4)‍

Check your understanding

Want to join the conversation?

Example: Finding the domain of $\frac{x + 1}{(x - 3) (x + 4)}$ ‍