Return to the Purplemath home page

 The Purplemath Forums
Helping students gain understanding
and self-confidence in algebra

powered by FreeFind


Return to the Lessons Index  | Do the Lessons in Order  |  Get "Purplemath on CD" for offline use  |  Print-friendly page

Dividing Rational Expressions (page 2 of 2)

For dividing rational expressions, you will use the same method as you used for dividing numerical fractions: when dividing by a fraction, you flip-n-multiply. For instance:

  • Perform the indicated operation:
    • (4/3)  (9/5)

    To simplify this division, I'll convert it to multiplication by flipping what I'm dividing by; that is, I'll switch from dividing by a fraction to multiplying by that fraction's reciprocal. Then I'll simplify as usual:

      (4/3)  (9/5) = (4/3)  (5/9) = 20/27

Can the 2's cancel off from the 20's? No! This is as simplified as the fraction gets.

Division works the same way with rational expressions.

  • Perform the indicated operation:
    • [ (x^2 + 2x - 15) / (x^2 - 4x - 45) ]  [ (x^2 + x - 12) / (x^2 - 5x - 36) ]

    To simplify this, first I'll flip-n-multiply. Then, to simplify the multiplication, I'll factor the numerators and denominators, and then cancel any duplicated factors. My work looks like this:

      [(x^2+2x-15)/(x^2-4x-45)]  [(x^2+x-12)/(x^2-5x-36)] = [(x^2+2x-15)/(x^2-4x-45)]  [(x^2-5x-36)/(x^2+x-12)] = [(x+5)(x-3)(x-9)(x+4)]/[(x-9)(x+5)(x+4)(x-3)] = 1

    Then the answer is: Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

      1, for x not equal to -5, -4, 3, or 9

Your instructor may not require the restrictions on the allowable x-values, in which case your answer would be just "1". The exercises you'll be given won't usually simplify this much, of course. The following example is much more typical:

  • Simplify the following expression:
    • [ (x^2 + 3x - 40) / (x^2 + 2x - 35) ]  [ (x^2 + 2x - 48) / (x^2 + 3x - 18) ]

    First, I'll need to flip the second fraction, and convert from division to multiplication. Then I'll factor, and see if anything cancels.

      [(x^2+3x-40)/(x^2+2x-35)]  [(x^2+2x-48)/(x^2+3x-18)] = [(x^2+3x-40)/(x^2+2x-35)]  [(x^2+3x-18)/(x^2+2x-48)] = [(x+8)(x-5)(x+6)(x-3)/(x+7)(x-5)(x+8)(x-6)] = [(x+6)(x-3)]/[(x+7)(x-6)]

(Can you cancel off the 6's? or the x's? No! The above is as simplified as this gets!)

    Then the final answer is:

      [ (x + 6)(x - 3) ] / [ (x + 7)(x - 6) ] = [x^2 + 3x - 18] / [ (x + 7)(x - 6) ] for x not equal to -8, -6, 3, or 5

For reasons which will become clear when adding and subtracting rationals, the numerator is usually multiplied out ("simplified" to get rid of the parentheses), while the denominator is usually left in factored form.

Make sure you know how to factor quadratics and cubics, because, as you have seen, it is required for many of the problems you'll be doing. Also, make sure you are careful to cancel only factors, not terms. If you can keep that straight, then you'll probably do fine.

<< Previous  Top  |  1 | 2  |  Return to Index

Cite this article as:

Stapel, Elizabeth. "Dividing Rational Expressions." Purplemath. Available from Accessed


  Linking to this site
  Printing pages
  School licensing

Reviews of
Internet Sites:
   Free Help
   Et Cetera

The "Homework

Study Skills Survey

Tutoring from Purplemath
Find a local math tutor

This lesson may be printed out for your personal use.

Content copyright protected by Copyscape website plagiarism search

  Copyright 2003-2012  Elizabeth Stapel   |   About   |   Terms of Use


 Feedback   |   Error?