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Homework Help: Help proving polynomial identity

  1. Apr 12, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove the following when p is a positive integer:
    [tex]b^p - a^p = (b-a)(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1})[/tex]

    Hint: Use the telescoping property for sums.

    2. Relevant equations
    None


    3. The attempt at a solution

    I tried reducing it to, [tex](b-a)\sum_{k=1}^p b^{p-k}a^{k-1}[/tex] but I wasn't able to do anything with it.


    I've been trying to work on this exercise which is a part of a problem set in induction. But I've been having quite a bit of difficulty and I'd appreciate it if somebody here could give me a hint as to the general direction. It's been bugging me for the past few days.
     
  2. jcsd
  3. Apr 12, 2010 #2

    epenguin

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    Homework Helper
    Gold Member

    I don't know what telescoping is but er

    [tex] (b-a)(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1})[/tex]

    is [tex]b(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1}) -a(b^{p-1}+b^{p-2}a+b^{p-3}a^2+...+ba^{p-2}+a^{p-1})[/tex].

    If you can't see what that gives, expand it out by executing the multiplications and collecting up terms.

    Quite a useful and important formula. When you do geometric series the same one is involved - make the relation.
     
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