# Rational Expressions: Finding the Domain

In this lesson, we'll look at the mathematical definition of a rational expression, and then move on to determining the domain that fulfills this expression. By the way, if you are not familiar with the lesson "Polynomials" and "Fractions" I advise you to first read them to understand the basics that we will need in this lesson.

A rational expression is the quotient of two polynomials. Mathematically;

### Note:

Let and two polynomials with . A rational expression is written as:

Examples of rational expression:

## Finding the Domain

As we said in the above property, the polynomial in the denominator must be different from 0. That is, the domain of the rational expression must not include solutions to the equation , so, we must exclude numbers from a rational expression’s domain that make the denominator zero. Because if the denominator is equal to 0 then the rational expression becomes undefined.

To determine the domain of the rational expression , we follow these two steps:

1. We solve the equation

2. We write the domain as the set of all real numbers excluding the solutions of the equation .

We can write the domain in two different ways:

1. Write the domain is all real numbers except ...

2. Using intervals

#### Example1:

Find the domain of the follwing rational expression:

#### Solution:

The denominator is and must be not equal to 0.

Solving the equation

Hence, for the denominator to be different from 0, must be different from 1.

The domain is "all real numbers except 1."

Or, with intervals:

#### Example2:

Find the domain of the follwing rational expression:

#### Solution:

The denominator is and must be not equal to 0.

Solving the equation

Hence, for the denominator to be different from 0, must be different from -2 and 5.

The domain is "all real numbers except -2 and 5."

Or, with intervals:

#### Example3:

Find the domain of the follwing rational expression:

#### Solution:

The denominator is and must be not equal to 0.

Solving the equation

Hence, for the denominator to be different from 0, must be different from 0 and 2.

The domain is "all real numbers except 0 and 2."

Or, with intervals:

#### Example4:

Find the domain of the follwing rational expression:

#### Solution:

Solving the equation

Hence, for the denominator to be different from 0, must be different from -1 and 2 and .

The domain is "all real numbers except -1 and 2 and ."

Or, with intervals:

#### Example5:

Find the domain of the follwing rational expression:

#### Solution:

To solve the equation , we can factorize the expression as

The solutions of the equation are -2 and 2.

Hence, for the denominator to be different from -2 and 2.

The domain is "all real numbers except -2 and 2"

Or, with intervals:

#### Example5:

Find the domain of the follwing rational expression:

#### Solution:

The equation is a quadratic equation, to solve it, we must calculate the quadratic formula.

So the equation has two solutions:

Hence, for the denominator to be different from -3 and 5.

The domain is "all real numbers except -3 and 5"

Or, with intervals:

#### Example6:

Find the domain of the follwing rational expression:

#### Solution:

The quadratic eqauation has no solution because whatever is, .

In this case, the domaine is "all real numbers"

Or with intervals,

#### Example7:

Find the domain of the follwing rational expression:

#### Solution:

The denominator is , and must be different from 0.

The domain of the rational expression is "all real numbers except 0".

With intervals, .