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Spivak, problem 1v

  1. Dec 19, 2007 #1
    Prove the following:

    [tex]x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})[/tex]

    Hmm ... I factored x then -y and came out with:

    [tex]x^{n}+...x^{2}y^{n-2}-x^{n-2}y^{2}...-y^{n}[/tex]

    Argh. What's up with the middle part? I'm not sure where to go from here.
     
    Last edited: Dec 19, 2007
  2. jcsd
  3. Dec 19, 2007 #2
    Direct expansion will verify it.
     
    Last edited: Dec 19, 2007
  4. Dec 19, 2007 #3
    Oops, I mistyped something.
     
  5. Dec 19, 2007 #4
    Putting the right hand side in sigma form

    [tex](x-y) \sum_{i=1}^{n} x^{n-i}y^{i-1}[/tex]

    [tex]\sum_{i=1}^{n} x^{n-i+1}y^{i-1} - x^{n-i}y^{i}[/tex]

    [tex]\sum_{i=0}^{n-1} x^{n-i}y^{i} - \sum_{i=1}^{n}x^{n-i}y^{i}[/tex]

    [tex]x^n - y^n + \sum_{i=1}^{n-1} x^{n-i}y^{i} - \sum_{i=1}^{n-1}x^{n-i}y^{i}[/tex]

    [tex]x^n - y^n[/tex]
     
  6. Dec 19, 2007 #5
    Too bad I have no idea what that means :-X I guess I'll just have to wait till I get to those types of methods. Thanks tho.
     
  7. Dec 19, 2007 #6

    morphism

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    Just expand the right side as mathboy suggested. chickendude just did it in abbreviated form.
     
  8. Dec 19, 2007 #7
    What does it mean to "expand" it. I think that's in the next chapter, so I'll just go back to it.
     
  9. Dec 19, 2007 #8

    morphism

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    It means to multiply the thing out, e.g. (x-y)(x+y) = x^2 + xy - yx - y^2. Have you done basic algebra? If not, I don't think Spivak is right for you just yet.
     
  10. Dec 19, 2007 #9
    Uh. That is exactly what I did smart ***. And what I got in the middle makes no sense to me.
     
  11. Dec 19, 2007 #10

    morphism

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    No reason to get snappy. :tongue2: Everything between x^n and -y^n will cancel off, e.g. we're going to get x^(n-1) y and -y x^(n-1), etc. Try it out for n=3 to get a feel for it.
     
  12. Feb 4, 2008 #11
    Hey,

    Sorry, for posting a little late. This problem is interesting out of what textbook did you get this problem from?

    Thanks,

    -PFStudent
     
  13. Feb 4, 2008 #12

    Defennder

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    It's Spivak's Calculus. He said so in the title.
     
  14. Feb 4, 2008 #13
    Hey,

    Ahem. Spivak (that is, Michael Spivak) is the author of several calculus titles,

    Calculus
    Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
    A Comprehensive Introduction to Differential Geometry, Volumes 1-4: 3rd Edition
    The Hitchhiker's Guide to Calculus
    Calculus: Calculus of Infinitesimals

    And no, he did not mention it was specifically from the text, Calculus.

    What I wanted to know was which one of his texts had the problem. It was already obvious that it was from one of his several calculus texts, however which one was not.

    Thanks,

    -PFStudent
     
    Last edited: Feb 4, 2008
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