7.2: Identify and Simplify Rational Expressions

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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 $\frac{4x^3}{12x^2}$ combined with techniques for factoring polynomials.

$Keep Calm and Be rational$

Keep Calm and be Rational

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

Evaluate $x=2$

Substitute $x=2$

$\begin{array}{l}\frac{2}{2-2}\\\text{}\\=\frac{2}{0}\end{array}$

Remember that you can’t divide by zero, so this means that for the expression $\frac{x}{x-2}$, 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 $x\neq{a}$

The reason you cannot divide any number c by zero $c \left( ?\,\,\cdot \,\,0\,\,=\,\,c \right)$. 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. $\frac{3x+2}{x-4}$
[reveal-answer q=”349619″]Show Solution[/reveal-answer]
[hidden-answer a=”349619″]

Find any values for x that would make the denominator equal 0.

$x–4=0$

When $x=4$, the denominator is equal to 0.

$x=4$

Answer

The domain is all real numbers, except 4.

Check

You found that $x\neq4$. Substitute that value into the expression to check that it gives an undefined mathematical operation.

$\begin{array}{c}\frac{3x+2}{x-4}\\\\\frac{3(4)+2}{(4)-4}\\\\\frac{12+2}{0}\\\\\frac{14}{0}\end{array}$

You find that when $x=4$, the numerator evaluates to 14, but the denominator evaluates to 0. And since division by 0 is undefined, this must be an excluded value.

[/hidden-answer]

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. $\frac{x+7}{{{x}^{2}}+8x-9}$
[reveal-answer q=”318517″]Show Solution[/reveal-answer]
[hidden-answer a=”318517″]

Find any values for x that would make the denominator equal to 0 by setting the denominator equal to 0 and solving the equation.

$x^{2}+8x-9=0$

Solve the equation by factoring. The solutions are the values that are excluded from the domain.

$\begin{array}{c}(x+9)(x-1)=0\\x=-9\,\,\,\text{or}\,\,\,x=1\end{array}$

Answer

The domain is all real numbers except $1$.

[/hidden-answer]

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:

$\frac{x^2+x-2}{x-1}$

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.

$\frac{5x^2}{10x}=\frac{5\cdot{x}\cdot{x}}{5\cdot{2}\cdot{x}}$

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

$\displaystyle\normalsize\frac{5\cdot{x}\cdot{x}}{5\cdot{2}\cdot{x}}=\frac{\cancel{5}\cdot{\cancel{x}}\cdot{x}}{\cancel{5}\cdot{2}\cdot{\cancel{x}}}=\frac{x}{2}$

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 $\frac{5x^{2}}{25x}$.

[reveal-answer q=”210856″]Show Solution[/reveal-answer]
[hidden-answer a=”210856″]Rewrite the numerator and denominator as factors.

$\frac{5x^{2}}{25x}=\frac{5\cdot{x}\cdot{x}}{5\cdot5\cdot{x}}$

Identify fractions that equal 1, and then simplify.

$\displaystyle\large\begin{array}{c}\frac{5\cdot{x}\cdot{x}}{5\cdot5\cdot{x}}\\\\=\frac{\cancel{5}\cdot{\cancel{x}}\cdot{x}}{\cancel{5}\cdot5\cdot{\cancel{x}}}\\\\=\frac{x}{5}\normalsize\cdot1\end{array}$

Simplify.

$\frac{x}{5}$

Answer

\(\frac{5

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}{25x}=\frac{x}{5}\)

[/hidden-answer]

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. $\frac{x+3}{{{x}^{2}}+12x+27}$
[reveal-answer q=”623785″]Show Solution[/reveal-answer]
[hidden-answer a=”623785″]

To find the domain (and the excluded values), find the values for which the denominator is equal to 0. Factor the quadratic, and apply the zero product principle.

$\begin{array}{c}x+3=0\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x+9=0\\x=0-3\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x=0-9\\x=-3\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x=-9\\\\x=-3\,\,\,\,\,\,\text{or}\,\,\,\,\,\,x=-9\end{array}$

The domain is all real numbers except $x=-9$.

Factor the numerator and denominator. Identify the factors that are the same in the numerator and denominator, and simplify.

$\large\begin{array}{c}\frac{x+3}{x^{2}+12x+27}\\\\=\frac{x+3}{\left(x+3\right)\left(x+9\right)}\\\\\frac{\cancel{x+3}}{\cancel{\left(x+3\right)}\left(x+9\right)}\\\\\normalsize=1\cdot\large\frac{1}{x+9}\end{array}$

Answer

$\frac{x+3}{{{x}^{2}}+12x+27}=\frac{1}{x+9}$

The domain is all real numbers except $−9$.

[/hidden-answer]

Example

Simplify and state the domain for the expression. $\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}$
[reveal-answer q=”861958″]Show Solution[/reveal-answer]
[hidden-answer a=”861958″]

To find the domain, determine the values for which the denominator is equal to 0.

$\begin{array}{r}x^{3}-x^{2}-20x=0\\x\left(x^{2}-x-20\right)=0\\x\left(x-5\right)\left(x+4\right)=0\end{array}$

The domain is all real numbers except 0, 5, and −4.

To simplify, factor the numerator and denominator of the rational expression. Identify the factors that are the same in the numerator and denominator, and simplify.

$\large\begin{array}{c}\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}\\\\=\frac{\left(x+4\right)\left(x+6\right)}{x\left(x-5\right)\left(x+4\right)}\\\\=\frac{\cancel{\left(x+4\right)}\left(x+6\right)}{x\left(x-5\right)\cancel{\left(x+4\right)}}\end{array}$

Simplify. It is acceptable to either leave the denominator in factored form or to distribute multiplication.

$\frac{x+6}{x\left(x-5\right)}\,\,\,\text{or}\,\,\,\frac{x+6}{x^{2}-5x}$

Answer

$\frac{x+6}{{{x}^{2}}-5x}$

The domain is all real numbers except 0, 5, and $−4$.

[/hidden-answer]

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

$Thumbnail for the embedded element "Simplify and Give the Domain of Rational Expressions"$

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=128

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.

Screenshot: Keep Calm. Provided by: Lumen Learning. License: Public Domain: No Known Copyright
Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
Simplify and Give the Domain of Rational Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/tJiz5rEktBs. License: CC BY: Attribution

Unit 15: Rational Expressions, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education. Located at: http://nrocnetwork.org/resources/downloads/nroc-math-open-textbook-units-1-12-pdf-and-word-formats/. License: CC BY: Attribution