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Difficult polynomial question involving factor and remainder theorems

  1. Mar 31, 2014 #1
    1. The problem statement, all variables and given/known data

    Prove that ##(a-b)## is a factor of ##a^5-b^5##, and find the other factor.

    2. Relevant equations

    Remainder theorem : remainder polynomial ##p(x)## divided by ##(x-a)## is equal to ##p(a)##
    Factor theorem : if remainder = 0, then divisor was a factor of dividend.

    3. The attempt at a solution

    I think am able to prove that it is a factor:

    ##P(x) = x^5 - b^5##; we replace a with x
    ##P(x) = (x-b)Q(x) + 0## ; we assume that (x-b) is a factor
    ##P(b) = (b-b)Q(x) + 0 = 0 ##; proves that .. um.. I think I'm going the wrong way anyway. This doesn't really prove anything? er.. eh.. ?

    And I can't really find the other factor :grumpy:
    I could do most of the other questions in the exercise, but not this one (and other related ones!)

    Any help would be appreciated
     
    Last edited: Mar 31, 2014
  2. jcsd
  3. Mar 31, 2014 #2

    SammyS

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    Do you know long division with polynomials, or synthetic division ?
     
    Last edited: Mar 31, 2014
  4. Mar 31, 2014 #3
    Yes. However, attempting polynomial division on (x^5+b^5) didn't quite work for me. If that's how to solve the question, could someone give me an example?
     
  5. Mar 31, 2014 #4

    Mark44

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  6. Apr 1, 2014 #5

    SteamKing

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    Maybe that's because you want to divide (a[itex]^{5}[/itex]-b[itex]^{5}[/itex]) by (a - b).

    Start off with a few warm-up exercises:

    (a[itex]^{2}[/itex]-b[itex]^{2}[/itex])/(a-b)

    (a[itex]^{3}[/itex]-b[itex]^{3}[/itex])/(a-b)

    You should know the answer to the first exercise by inspection.
     
  7. Apr 1, 2014 #6
    Woops, got what I missed. Thanks!
     
  8. Apr 1, 2014 #7

    epenguin

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