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Polynomial Long division

  1. Jan 22, 2015 #1
    1. The problem statement, all variables and given/known data
    [tex]\frac{x^5-a^5}{x^2-a^2}[/tex], where a is some constant.

    2. Relevant equations


    3. The attempt at a solution
    I can't figure out how to do this with long division. With synthetic, I can get to [tex]\frac{a^4+a^3 x+a^2 x^2+a x^3+x^4}{a+x}[/tex]



    Code (Text):

                      x^3+xa^2+?
                    _______________
       x^2-a^2      ) x^5-a^5
                          -x^5 + x^3a^2
                        ---------------
                              0-  a^5+x^3a^2
                                    xa^4-x^3a^2
                        ---------------
                                     
    Hopefully it's possible to decipher my steps from that diagram, I don't know how to write long division in latex. I'm left with -a^5+xa^4, which doesn't go evenly into the divisor. I thought about writing [tex]\frac{-a^5+xa^4}{x^2-a^2}[/tex] in place of the ? mark, but the entire purpose of this is to take the limit as x->a, and I would be left with division by zero. Surely this isn't only solvable by synthetic division?

    Edit: For prosperity's sake, the limit is (5a^3)/2
     
  2. jcsd
  3. Jan 22, 2015 #2

    vela

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    What you did so far is fine. You can simplify your expression for the remainder. Factor ##a^4## out of the numerator, and factor the denominator. You'll get some cancellation that will allows you to evaluate the limit.
     
  4. Jan 22, 2015 #3
    I'm not seeing the cancellation. Could you elaborate?
     
  5. Jan 22, 2015 #4

    vela

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    Not really without pretty much doing it for you. What did you get when you factored the top and bottom?
     
  6. Jan 22, 2015 #5
    Sorry, I forgot about the -a^5. I got it now, thanks!
     
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