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).

      (29 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)

    • Kim Seidel

      6 years agoPosted 6 years ago. Direct link to Kim Seidel's post “The denominator is: x^2+4...”

      Intro to rational expressions (article) | Khan Academy (9)

      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.

  • 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

    10 months agoPosted 10 months ago. Direct link to jher4900's post “what is the equation for ...”

    what is the equation for a rational function?

    (1 vote)

    • Kim Seidel

      10 months agoPosted 10 months 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.

      EDIT

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

      (2 votes)

  • Yong Bakos

    a year agoPosted a year 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)

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