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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, squared, minus, 6, x, minus, 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:
  • start fraction, 1, divided by, x, end fraction
  • start fraction, x, plus, 5, divided by, x, squared, minus, 4, x, plus, 4, end fraction
  • start fraction, x, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, 2, x, minus, 3, right parenthesis, divided by, x, minus, 6, end fraction
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 start fraction, 2, x, plus, 3, divided by, x, minus, 2, end fraction.
We can determine the value of this expression for particular x-values. For example, let's evaluate the expression at start color #11accd, x, end color #11accd, equals, start color #11accd, 1, end color #11accd.
2(1)+312=51=5\begin{aligned} \dfrac{2(\blueD{1})+3}{\blueD1-2} &= \dfrac{5}{-1} \\\\ &=\goldD{-5} \end{aligned}
From this, we see that the value of the expression at start color #11accd, x, end color #11accd, equals, start color #11accd, 1, end color #11accd is start color #e07d10, minus, 5, end color #e07d10.
Now let's find the value of the expression at start color #11accd, x, end color #11accd, equals, start color #11accd, 2, end color #11accd.
2(2)+322=70=undefined!\begin{aligned} \dfrac{2(\blueD{2})+3}{\blueD2-2} &= \dfrac{7}{0} \\\\ &=\goldD{\text{undefined!}} \end{aligned}
An input of 2 makes the denominator 0. Since division by 0 is undefined, start color #11accd, x, end color #11accd, equals, start color #11accd, 2, end color #11accd 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 start fraction, x, plus, 1, divided by, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 4, right parenthesis, end fraction

Let's find the zeros of the denominator and then restrict these values:
(x3)(x+4)=0x3=0orx+4=0x=3orx=4\begin{aligned} (x-3)&(x+4)= 0 \\\\ \swarrow \quad&\quad \searrow \\\\ x-3=0 \quad &\text{or} \quad x+4=0 \\\\ x = 3 \quad&\text{or} \quad x=-4 \end{aligned}
So we write that the domain is all real numbers except start text, 3, end text and start text, negative, 4, end text, or simply x, does not equal, 3, comma, minus, 4.

Check your understanding

Problem 1
What is the domain of start fraction, x, plus, 1, divided by, x, minus, 7, end fraction?
Choose 1 answer:
Choose 1 answer:

Problem 2
What is the domain of start fraction, 3, x, minus, 7, divided by, 2, x, plus, 1, end fraction?
Choose 1 answer:
Choose 1 answer:

Problem 3
What is the domain of start fraction, 2, x, minus, 3, divided by, x, left parenthesis, x, plus, 1, right parenthesis, end fraction?
Choose 1 answer:
Choose 1 answer:

Problem 4
What is the domain of start fraction, x, minus, 3, divided by, x, squared, minus, 2, x, minus, 8, end fraction?
Choose 1 answer:
Choose 1 answer:

Problem 5
What is the domain of start fraction, x, plus, 2, divided by, x, squared, plus, 4, end fraction?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • duskpin sapling style avatar for user Jenn Vo
    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!
    (22 votes)
    Default Khan Academy avatar avatar for user
    • primosaur ultimate style avatar for user Mr. Brownridge
      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).
      (18 votes)
  • blobby green style avatar for user MatthewS
    I don't have a good understanding of how exactly you find the domain, and what "all real numbers" means.
    (6 votes)
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    • duskpin ultimate style avatar for user Victor
      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!
      (9 votes)
  • duskpin sapling style avatar for user aaliyahmariecerveny
    Why is number 5, all real numbers shouldn't it be +/- 2 since x^2=+4, factors out to (x+2)(x-2)?
    (7 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (1 vote)
  • blobby green style avatar for user Mrs. Head
    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).
    (4 votes)
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  • leaf orange style avatar for user hwang
    In rational expression why is domain all real number?
    (3 votes)
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    • female robot grace style avatar for user loumast17
      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.
      (2 votes)
  • blobby green style avatar for user Fred Haynes
    In problem # 3, the denominator is x(x+1). Why can't the zero simply be -1? Because -1+1 =0 and x*0=0.

    Thanks in advance.
    (3 votes)
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    • duskpin sapling style avatar for user Elissa
      If x was just -1, what if you got an answer of 0? Then your denominator would be 0 and you can't have a denominator of 0. You need to write it out so that you know that, if you get an answer of 0, you have a wrong answer.
      (1 vote)
  • leaf green style avatar for user Yu Aoi
    if a constant is a polynomial, is 1/2 a rational expression?
    (2 votes)
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  • blobby green style avatar for user Annalishia Johnson
    is it bad that Im just starting to understand this subject
    (2 votes)
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  • marcimus pink style avatar for user kaplan.sophie4
    In the third paragraph of this article, the text describes a rational expression as a "ratio of two polynomials. Or in other words, it is a fraction", thereby implying that a ratio and fraction are the same. However, I have learned from some teachers that a ratio is not to be confused with a fraction. A ratio, as Khan Academy states, is a comparison of two quantities while a fraction is a number that names part of a whole or part of a group. A ratio of 3:4 would describe that there are three of one thing and four of the other. A fraction of 3/4 would describe having three of the four things. So isn't a rational expression only a fraction?
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      When talking about types of numbers, the 2 terms (ratios and fractions) are used a little more loosely...
      The definition of rational numbers is that a rational number is a number that can be written as a ratio of 2 integers. Any fraction where the numerator and denominator are integers fits this definition. In fact, you will usually hear fractions referred to as rational numbers and vice versa. Simplified ratios can also fit this definition.

      Sal is extending this definition into rational expressions.

      Hope this helps.
      (0 votes)
  • blobby green style avatar for user Jad El-rabaa
    how would i know if they are all real numbers ? that really confuses me
    (2 votes)
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    • leaf orange style avatar for user A/V
      Real numbers are any and all numbers on a number line. Anything in between -inf<x<inf is a real number. The only time when you should not characterize a number as a real number is when it has an imaginary number, i.
      (2 votes)