# 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

1. $\dfrac{x-1}{x+2}$

2. $\dfrac{x+1}{(x+2)(x-3)}$

3. $\dfrac{1}{x^2 + 1}$

4. $\dfrac{x^2+4}{x^3+2 x^2-3x}$

## 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 $\dfrac{x-1}{x+2}$

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

2. The domain of the rational expression $\dfrac{x+1}{(x+2)(x-3)}$

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

3. The domain of the rational expression $\dfrac{1}{x^2+1}$

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, $\dfrac{x^2+4}{x^3+2x^2-3x}$

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

$x^3 + 2 x^2 - 3 x = x(x^2 + 2 x - 3)$

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

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.

1. $\dfrac{x+9}{x-10}$

2. $\dfrac{1}{(x-9)(x+1)}$

3. $\dfrac{1}{x^3+1}$

4. $\dfrac{-x+7}{x^4-16}$

More references to topics related to rational expressions.

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Updated: 2 April 2013