### Learning Objectives

- Recognize and define a rational expression
- Determine the domain of a rational expression
- Simplify a rational expression

**Rational expressions** are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified using the techniques used to simplify expressions such as [latex]\frac{4x^3}{12x^2}[/latex] combined with techniques for factoring polynomials.

To introduce the first important feature of rational expressions and equations, we will show an example.

Evaluate [latex]\frac{x}{x-2}[/latex] for [latex]x=2[/latex]

Substitute [latex]x=2[/latex]

[latex]\begin{array}{l}\frac{2}{2-2}\\\text{}\\=\frac{2}{0}\end{array}[/latex]

Remember that you can’t divide by zero, so this means that for the expression [latex]\frac{x}{x-2}[/latex], x cannot be 2 because it will result in an undefined ratio. In general, finding values for a variable that will not result in division by zero is called finding the domain.

#### Domain of a rational expression or equation

The domain of a rational expression or equation is a collection of the values for the variable that will not result in an undefined mathematical operation such as division by zero. For a = any real number, we can notate the domain in the following way:

x is all real numbers where [latex]x\neq{a}[/latex]

The reason you cannot divide any number *c* by zero [latex] \left( \frac{c}{0}\,\,=\,\,? \right)[/latex] is that you would have to find a number that when you multiply it by 0 you would get back [latex]c \left( ?\,\,\cdot \,\,0\,\,=\,\,c \right)[/latex]. There are no numbers that can do this, so we say “division by zero is undefined”. In simplifying rational expressions you need to pay attention to what values of the variable(s) in the expression would make the denominator equal zero. These values cannot be included in the domain, so they’re called excluded values. Discard them right at the start, before you go any further.

(Note that although the *denominator* cannot be equivalent to 0, the *numerator* can—this is why you only look for excluded values in the denominator of a rational expression.)

For rational expressions, the domain will exclude values for which the value of the denominator is 0. The following two examples illustrate finding the domain of an expression.

### Example

Identify the domain of the expression. [latex] \frac{3x+2}{x-4}[/latex]

In the next example we will identify the domain of a rational expression that contains polynomials in the numerator and denominator.

### Example

Identify the domain of the expression. [latex] \frac{x+7}{{{x}^{2}}+8x-9}[/latex]

As with many other mathematical expressions and equations, it can be very helpful to simplify rational expressions. We simplified rational expressions with monomial terms in the exponents module. Here we will combine what we know about factoring polynomials with factoring rational expressions that have monomial terms. The goal is to be able to simplify an expression such as this:

[latex]\frac{x^2+x-2}{x-1}[/latex]

Before we offer more examples, we will show you a technique for simplifying rational expressions that will make things a bit easier. The idea is that a number or variable divided by itself is equal to one, so we can factor a rational expression and identify common factors between the numerator and denominator.

[latex]\frac{5x^2}{10x}=\frac{5\cdot{x}\cdot{x}}{5\cdot{2}\cdot{x}}[/latex]

The common factors between the numerator and denominator are 5 and x, so we can “cancel” them to show that [latex]\frac{5}{5}=1\text{ and }\frac{x}{x}=1[/latex].

[latex]\displaystyle\normalsize\frac{5\cdot{x}\cdot{x}}{5\cdot{2}\cdot{x}}=\frac{\cancel{5}\cdot{\cancel{x}}\cdot{x}}{\cancel{5}\cdot{2}\cdot{\cancel{x}}}=\frac{x}{2}[/latex]

The next example provides a reminder of how to simplify a monomial with variables and exponents. We will then use this idea to simplify a rational expression and define it’s domain.

### Example

Simplify [latex]\frac{5x^{2}}{25x}[/latex].

We can summarize the process as follows: Factor the numerator, factor the denominator, identify factors that are common to the numerator and denominator, cancel them to represent division, and simplify.

When simplifying rational expressions, it is a good habit to always consider the domain first. This will come in handy when you begin *solving* rational equations a bit later on. When finding the domain of an expression, you always start with the original expression because variable terms may be factored out as part of the simplification process.

In the examples that follow, the numerator and the denominator are polynomials with more than one term, but the same principles of simplifying will once again apply. Factor the numerator and denominator to simplify the rational expression.

### Example

Simplify and state the domain for the expression. [latex] \frac{x+3}{{{x}^{2}}+12x+27}[/latex]

### Example

Simplify and state the domain for the expression. [latex]\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}[/latex]

In the following video we present another example of finding the domain of a rational expression.

### Steps for Simplifying a Rational Expression

To simplify a rational expression, follow these steps:

- Determine the domain. The excluded values are those values for the variable that result in the expression having a denominator of 0.
- Factor the numerator and denominator.
- Find common factors for the numerator and denominator and simplify.

## Summary

An additional consideration for rational expressions is to determine what values are excluded from the domain. Since division by 0 is undefined, any values of the variables that result in a denominator of 0 must be excluded. Excluded values must be identified in the original equation, not from its factored form.Rational expressions are fractions containing polynomials. They can be simplified much like numeric fractions. To simplify a rational expression, first determine common factors of the numerator and denominator, and then remove them by rewriting them as expressions equal to 1.