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



Rational Expressions: More Simplifying (page 3 of 3)

Sections: Finding the domain, Simplifying rational expressions


  • Simplify the following rational expression:
    • [ 2x^2 + 13x + 20 ] / [ 2x^2 + 17x + 30 ]

    Many students will try to do something like the following:

      This is just SO wrong...

    Is this legitimate? Can the student really do this? (Think "bleeding, oozing...") You can NOT cancel term-by-term! You can ONLY cancel factors!

    So the first thing I have to do (if I'm going to do the simplification correctly) is factor the numerator and the denominator:

      [ 2x^2 + 13x + 20 ] / [ 2x^2 + 17x + 30 ] = [ (2x + 5) (x + 4) ] / [ (x + 6) (2x + 5) ]

    Since the numerator and denominator share a common factor, I can reduce the expression as:

      [ 2x^2 + 13x + 20 ] / [ 2x^2 + 17x + 30 ] = (x + 4) / (x + 6)

    Can I reduce any further? For instance, can I cancel off the x's? (whimpering, bleeding...) Can I cancel a 2 out of the 4 and the 6? (oozing, flopping...) No! This is as simplified as it's going to get, because there are no remaining common factors. Then the answer is:

      (x + 4) / (x + 6) for x not equal to -5/2

Depending on your text, you might not need that "for x not equal to 5/2 part". However, since I cancelled off a "2x + 5" factor, this removed a division-by-zero problem from the original rational expression: 2x + 5 = 0 for x = 5/2.

  • Simplify the following:
    • (x - 2) / (2 - x)

    The factors in the numerator and denominator are almost the same, but not quite, so they can't be cancelled yet. If the fraction had been: Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

     

    ADVERTISEMENT

     

      (x + 2) / (2 + x)

    ...(that is, plusses instead of minuses), I could have rearranged the terms as:

      (x + 2) / (2 + x) = (x + 2) / (x + 2)

    ...and cancelled to get "1", since order doesn't matter in addition. But order most-definitely does matter in subtraction, so I can't just flip the subtraction to get matching factors. However, take a look at this:

      5 3 = 2
      3 5 = 2

    Do you see what happened? When I reversed the subtraction in the second line, I got the same number but the opposite sign. Then, if I flip a subtraction, I'll need to change the sign. So I can reverse one of the subtractions in the original rational expression above, as long as I remember to switch the sign out front:

      (x - 2) / (2 - x) = (x - 2) / [ -(x - 2) ]

    Now I can cancel:

      (x - 2) / (2 - x) = (x - 2) / [ -(x - 2) ] = (x - 2) / [ -1(x - 2) ]

    Remember: If "nothing" is left, then a "1" is left, so:

      (x - 2) / (2 - x) = -1

(Depending on the text you're using, you may or may not need the "as long as x does not equal 2" part.)

You should keep this "flip a subtraction and kick a 'minus' sign out front" trick in mind. Depending on the text you're using, you may see a lot of this.

  • Simplify the following expression:
    • (x + 3) / (x^2 - x - 12)

    To simplify this, I first need to factor. Then I can cancel off any common factors.

      (x + 3) / (x^2 - x - 12) = (x + 3) / [ (x + 3) (x - 4) ] = 1/(x - 4)

    Then the answer is:

      1/(x - 4) for x not equal to - 3

(You might not need the "for all x not equal to 3" part.)

  • Simplify the following expression:
    • (x^2 - 36) / (6 - x)

    To simplify this, I need to factor; I'll also need to flip the subtraction in the denominator, so I'll need to remember to change the sign.

      (x^2 - 36) / (6 - x) = [ (x - 6)(x + 6) ] / [ -1(x - 6) ] = -(x + 6 = -x - 6

    Then the answer is:

      -x - 6 for x not equal to 6

(You might not need the "for all x not equal to 6" part.)

As you have probably noticed by now, simplifying rational expressions involves a lot of factoring. If you're feeling at all rusty on this topic, review now: simple factoring, factoring quadratics, and special factoring formulas.

<< Previous  Top  |  1 | 2 | 3  |  Return to Index

Cite this article as:

Stapel, Elizabeth. "Rational Expressions: More Simplifying." Purplemath. Available from
    http://www.purplemath.com/modules/rtnldefs3.htm. Accessed
 

 



Purplemath:
  Linking to this site
  Printing pages
  School licensing


Reviews of
Internet Sites:
   Free Help
   Practice
   Et Cetera

The "Homework
   Guidelines"

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?