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### Course: College Algebra>Unit 8

Lesson 3: Reducing rational expressions to lowest terms

# Intro to rational expressions

Learn what rational expressions are and about the values for which they are undefined.

#### What you will learn in this lesson

This lesson will introduce you to rational expressions. You will learn how to determine when a rational expression is undefined and how to find its domain.

## What is a rational expression?

A polynomial is an expression that consists of a sum of terms containing integer powers of $x$, like $3{x}^{2}-6x-1$.
A rational expression is simply a quotient of two polynomials. Or in other words, it is a fraction whose numerator and denominator are polynomials.
These are examples of rational expressions:
• $\frac{1}{x}$
• $\frac{x+5}{{x}^{2}-4x+4}$
• $\frac{x\left(x+1\right)\left(2x-3\right)}{x-6}$
Notice that the numerator can be a constant and that the polynomials can be of varying degrees and in multiple forms.

## Rational expressions and undefined values

Consider the rational expression $\frac{2x+3}{x-2}$.
We can determine the value of this expression for particular $x$-values. For example, let's evaluate the expression at $x=1$.
$\begin{array}{rl}\frac{2\left(1\right)+3}{1-2}& =\frac{5}{-1}\\ \\ & =-5\end{array}$
From this, we see that the value of the expression at $x=1$ is $-5$.
Now let's find the value of the expression at $x=2$.
$\begin{array}{rl}\frac{2\left(2\right)+3}{2-2}& =\frac{7}{0}\\ \\ & =\text{undefined!}\end{array}$
An input of $2$ makes the denominator $0$. Since division by $0$ is undefined, $x=2$ is not a possible input for this expression!

## Domain of rational expressions

The domain of any expression is the set of all possible input values.
In the case of rational expressions, we can input any value except for those that make the denominator equal to $0$ (since division by $0$ is undefined).
In other words, the domain of a rational expression includes all real numbers except for those that make its denominator zero.

### Example: Finding the domain of $\frac{x+1}{\left(x-3\right)\left(x+4\right)}$‍

Let's find the zeros of the denominator and then restrict these values:
$\begin{array}{rl}\left(x-3\right)& \left(x+4\right)=0\\ \\ ↙\phantom{\rule{1em}{0ex}}& \phantom{\rule{1em}{0ex}}↘\\ \\ x-3=0\phantom{\rule{1em}{0ex}}& \text{or}\phantom{\rule{1em}{0ex}}x+4=0\\ \\ x=3\phantom{\rule{1em}{0ex}}& \text{or}\phantom{\rule{1em}{0ex}}x=-4\end{array}$
So we write that the domain is all real numbers except $\mathit{\text{3}}$ and $\mathit{\text{-4}}$, or simply $x\ne 3,-4$.

Problem 1
What is the domain of $\frac{x+1}{x-7}$?

Problem 2
What is the domain of $\frac{3x-7}{2x+1}$?

Problem 3
What is the domain of $\frac{2x-3}{x\left(x+1\right)}$?

Problem 4
What is the domain of $\frac{x-3}{{x}^{2}-2x-8}$?

Problem 5
What is the domain of $\frac{x+2}{{x}^{2}+4}$?

## Want to join the conversation?

• I have a question about #5 under the Check your understanding section. So for the denominator in that fraction, can I use the method "the different of 2 squares" to factor it out to (x+2) (x-2) and solve for x from there? Can you explain more about it? I didn't get the last part in the explanation. Thanks!
• Difference refers to subtraction. x^2+4 is a sum. Therefore, it is a "sum of two squares." If you graph the function you will see that it is an upward facing parabola with a y-intercept of 4. It has no solutions. So the expression will never equal zero (unless we use a different set of numbers called complex numbers).
• Why is number 5, all real numbers shouldn't it be +/- 2 since x^2=+4, factors out to (x+2)(x-2)?
• The denominator is: x^2+4. You changed it into x^2-4.
x^2+4 is not factorable. Any real number squared will create a positive value. That positive value plus 4 creates an even larger positive value. There is no value that you can use for X that would cause the denominator to become 0. This is why the answer is that the domain = all real numbers.

Hope this helps.
• For Problem 5, why can’t x= +- 2i ?
• We define domain and range using the set of real numbers. The domain in problem 5 is all real numbers. There is no value of x that makes the denominator = 0, so there are no values to exclude from the domain.

You are asking about imaginary numbers. They are outside the set of real numbers, so they are no considered.

Hope this helps.
• I don't have a good understanding of how exactly you find the domain, and what "all real numbers" means.
• Domain means that you are trying to find all possible values of x. Domain's are usually written in this format: {xeR} where xeR means that for every real number, x is a solution. All real numbers mean any number that exists, and they may be irrational, rational, negative, positive, etc. However, they cannot be undefinable values such as √-1, which is i in short. In order to find the domain, you'll have to find what can't be in the denominator usually by factoring, and you'll be able to find out what x cannot be. If you have a specific question you'd like me to walk you through, don't hesitate to ask!
• explain why domain of a rational expression is all real numbers except for those that make the denominator equal to zero.
• When the denominator is 0, you are dividing by 0. Division by 0 is undefined, so any values that cause that are not included in the domain.

Otherwise, you can divide by any other number as long as it isn’t 0.
• Why do you use the term "cancel"? I know a lot of teachers use it and that was what my teachers called it when I was in school. But is this really a mathematically correct term?
I spend a great deal of time correcting students who just want to "cancel" terms just because they are alike, without understanding that in order for terms to be removed from an expression you have to use a mathematical operation, division or subtraction. Therefore terms can only be "divided out" or "subtracted out". Students will often times cross out or as you say "cancel out" terms that are both in numerators when multiplying terms or both in the denominators. To help resolve this issue my students are only allowed to use correct mathematical operations when simplifying expressions (divide out or subtract out).
• what is the equation for a rational function?
(1 vote)
• There is no single equation for rational functions. Any function that involves fractions would be a rational function.
• In rational expression why is domain all real number?
• rational expressions depend on the denominator for domain. If you know how to find vertical asymptotes and holes, those are what would limit the domain of a rational function. The only time a rational function has a domain of all reals is if the denominator is just 1.

EDIT

Thanks to Hecretary Bird for his correction. denominator just has to be a constant, other than 0 still though.