# Rational Expressions

The definition of rational expressions and their domains are examined.

## Definition

A rational expression is the quotient of two polynomials.

## Examples of Rational Expressions ## Domain of Rational Expressions

It is clear that the above expressions are undefined if a division by 0 occurs. The domain of a rational expression excludes all values that make the denominator equal to 0.
1. The domain of the rational expression (x - 1) / (x + 2)

is the set of all real numbers except x = - 2

2. The domain of the rational expression (x + 1) / ((x + 2)(x - 3))

is the set of all real numbers except x = -2 and x = 3

3. The domain of the rational expression 1 / (x2 + 1)

is the set of all real numbers since the denominator x2 + 1 cannot be 0 for any real number x.

4. To find the domain of the rational expression, (x2 + 4) / (x3 + 2 x2 - 3 x)

we first have to factor the denominator and find its zeros.

x3 + 2 x2 - 3 x = x(x2 + 2 x - 3)

= x(x - 1)(x + 3)

The domain of the above rational expressions is the set of all real numbers except x = 0, x = 1 and x = - 3.

## Exercises

Find the domain of each of the Rational Expressions given below. More references to topics related to rational expressions.