Intro to rational expressions (article) | Khan Academy (2024)

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

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  • Jenn Vo

    8 years agoPosted 8 years ago. Direct link to Jenn Vo's post “I have a question about #...”

    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!

    (25 votes)

    • Mr. Brownridge

      8 years agoPosted 8 years ago. Direct link to Mr. Brownridge's post “Difference refers to subt...”

      Intro to rational expressions (article) | Khan Academy (4)

      Intro to rational expressions (article) | Khan Academy (5)

      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).

      (30 votes)

  • aaliyahmariecerveny

    6 years agoPosted 6 years ago. Direct link to aaliyahmariecerveny's post “Why is number 5, _all rea...”

    Why is number 5, all real numbers shouldn't it be +/- 2 since x^2=+4, factors out to (x+2)(x-2)?

    (8 votes)

  • Vishwa Patel

    a year agoPosted a year ago. Direct link to Vishwa Patel's post “For Problem 5, why can’t ...”

    For Problem 5, why can’t x= +- 2i ?

    (5 votes)

    • Kim Seidel

      a year agoPosted a year ago. Direct link to Kim Seidel's post “We define domain and rang...”

      Intro to rational expressions (article) | Khan Academy (13)

      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.

      (15 votes)

  • MatthewS

    4 years agoPosted 4 years ago. Direct link to MatthewS's post “I don't have a good under...”

    I don't have a good understanding of how exactly you find the domain, and what "all real numbers" means.

    (6 votes)

    • Victor

      4 years agoPosted 4 years ago. Direct link to Victor's post “Domain means that you are...”

      Intro to rational expressions (article) | Khan Academy (17)

      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!

      (10 votes)

  • zunnunam

    a year agoPosted a year ago. Direct link to zunnunam's post “explain why domain of a r...”

    explain why domain of a rational expression is all real numbers except for those that make the denominator equal to zero.

    (4 votes)

    • Tanner P

      a year agoPosted a year ago. Direct link to Tanner P's post “When the denominator is 0...”

      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.

      (8 votes)

  • Mrs. Head

    4 years agoPosted 4 years ago. Direct link to Mrs. Head's post “Why do you use the term "...”

    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).

    (5 votes)

  • jher4900

    a year agoPosted a year ago. Direct link to jher4900's post “what is the equation for ...”

    what is the equation for a rational function?

    (1 vote)

    • Kim Seidel

      a year agoPosted a year ago. Direct link to Kim Seidel's post “There is no single equati...”

      There is no single equation for rational functions. Any function that involves fractions would be a rational function.

      (6 votes)

  • hwang

    4 years agoPosted 4 years ago. Direct link to hwang's post “In rational expression wh...”

    In rational expression why is domain all real number?

    (3 votes)

    • loumast17

      4 years agoPosted 4 years ago. Direct link to loumast17's post “rational expressions depe...”

      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.


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

      (2 votes)

  • Yong Bakos

    2 years agoPosted 2 years ago. Direct link to Yong Bakos's post “Do we have to be mindful ...”

    Do we have to be mindful of the domain during intermediate steps of equation solving? For example if I have an equation and divide both sides by x, do I have to state that, in my final solution of the equation, that x cannot be 0?

    (3 votes)

    • Robin

      a year agoPosted a year ago. Direct link to Robin's post “I'm pretty sure?”

      I'm pretty sure?

      (2 votes)

  • nestor.mendez

    2 years agoPosted 2 years ago. Direct link to nestor.mendez's post “how do I know what is and...”

    how do I know what is and isn't a real number you didn't really explain

    (2 votes)

    • rainpaw10

      a year agoPosted a year ago. Direct link to rainpaw10's post “There are other topics ab...”

      There are other topics about this on Khan Academy that can explain it better, but basically, a real number is any number that is not an imaginary number like i. Pretty much any number that you can think of is a real number!

      (4 votes)

Intro to rational expressions (article) | Khan Academy (2024)


How to know if an expression is rational? ›

Definitions: A rational expression is the ratio of two polynomials. If f is a rational expression then f can be written in the form p/q where p and q are polynomials.

How to add and subtract rational expressions? ›

To add or subtract two rational expressions with the same denominator, we simply add or subtract the numerators and write the result over the common denominator. When the denominators are not the same, we must manipulate them so that they become the same.

How do you write a rational expression? ›

In a rational expression, both numerator and denominator are polynomials. i.e., it is of the form p(x)/q(x), where q(x) ≠ 0 and p(x) and q(x) are polynomials. Since rational expressions are nothing but fractions, we operate on them just the way we operate the fractions.

How do you simplify rational expressions step by step? ›

Step 1: Factor the numerator and the denominator. Step 2: List restricted values. Step 3: Cancel common factors. Step 4: Reduce to lowest terms and note any restricted values not implied by the expression.

How do you solve rational equations for dummies? ›

By breaking down this approach into the following steps you can solve scary-looking rational equations like this one:
  1. Find a common denominator for all the terms in the equation. ...
  2. Write each fraction with the common denominator. ...
  3. Multiply each side of the equation by that same denominator. ...
  4. Solve the new equation.
Mar 26, 2016

What are the rules for rational expressions? ›

Rules of Rational Expressions
  • Rule 1: When the two expressions with bases are multiplied, the powers are added together.
  • Rule 2: When the two expressions with bases are divided, the powers are subtracted.
  • Rule 4: For an exponent function with negative power, the result is the reciprocal of the original function.
May 3, 2023

What is the formula of a rational equation? ›

A rational equation is an equation that contains fractions with xs in the numerator, denominator or both. Here is an example of a rational equation: (4 / (x + 1)) - (3 / (x - 1)) = -2 / (x^2 - 1).

Is 0 a rational number? ›

Yes, zero is a rational number.

A rational no. is a number represented as p/q, where q and p are integers and q ≠ 0. This States that 0 is a rational number because any number can be divided by 0 and equal 0.

How to divide rational expressions? ›

When we divide rational expressions, we multiply the dividend (the first expression) by the reciprocal of the divisor (the second expression). We can also see if we can reduce the quotient to lowest terms. This is very similar to dividing fractions, only we also have to think about the domain while we do it.

How do you rewrite rational expressions to find sums and differences? ›

Adding or subtracting rational expressions is a four-step process:
  1. Write all fractions as equivalent fractions with a common denominator.
  2. Combine the fractions as a single fraction that has the common denominator.
  3. Simplify the expression in the top of the fraction.
  4. Reduce the fraction to lowest terms.

What are the common mistakes in simplifying rational expressions? ›

A common error when simplifying rational functions includes failing to factor completely before canceling out common factors in the numerator and denominator.

What is the simplest form of a rational expression? ›

A rational expression is considered simplified if there are no common factors in its numerator and denominator. For example: 2 3 2 3 is simplified because there are no common factors of 2 and 3.

How to tell if a rational expression is proper or improper? ›

Like fractions, rational expressions can be proper or improper. A proper rational expression has a lower-degree numerator than denominator ( e . g . , 1 − x / x 2 + 3 ) , and an improper one has a higher-degree numerator than denominator ( e . g . , x 2 + 3 / 1 − x ) .

What is the formula for a rational function? ›

A rational function equation is of the form f(x) = P(x) / Q(x), where Q(x) ≠ 0. Every rational function has at least one vertical asymptote. Every rational function has at most one horizontal asymptote. Every rational function has at most one slant asymptote.

What are 5 examples of rational equations? ›

This is the 5 examples of rational equation:
  • 4/x + 5/2 = -11/x.
  • 5x/(x - 2) = 7 + 10/(x - 2)
  • ( 3x - 2)/(x - 2) = 6/(x2 - 4) + 1.
  • 2/(x2 - x) = 1/(x - 1)
  • 3/(x + 2) = 6/(x - 1)
Nov 8, 2020

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