Polynomial Long division

  • #1

Homework Statement


[tex]\frac{x^5-a^5}{x^2-a^2}[/tex], where a is some constant.

Homework Equations




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:
                  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
 

Answers and Replies

  • #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.
 
  • #3
I'm not seeing the cancellation. Could you elaborate?
 
  • #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?
 
  • #5
Sorry, I forgot about the -a^5. I got it now, thanks!
 

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