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X^n +y^n

  1. Mar 18, 2010 #1
    What is the expansion of xn +yn , when is even ??/
     
  2. jcsd
  3. Mar 18, 2010 #2
    I don't see anything that can be expanded.
     
  4. Mar 18, 2010 #3
    i mean that xn - yn can be written as (x-y)(xn-1 +xn-2y ....+yn-1 )
    similarly what can xn +yn be written as ????????
     
  5. Mar 18, 2010 #4

    jav

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    Try alternating signs, and it becomes straightforward.
     
  6. Mar 18, 2010 #5
    I think you need to think about zeros
    [tex]x^n+y^n=0[/tex]
    [tex]x^n=-y^n[/tex]
    [tex]x=y\cdot\exp(i\pi k/n)[/tex]
    [tex]\therefore x^n+y^n=\prod_k (x-\exp(i\pi k/n)y)[/tex]

    Occationally combining a subset of these factors together will give you a real solution.

    Now you need to think when... :)
     
    Last edited: Mar 18, 2010
  7. Mar 18, 2010 #6

    Mentallic

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    Homework Helper

    Then you mean what are the factors :smile:

    If n is odd, you can factor it as so:

    [tex]x^n+y^n=(x+y)(x^{n-1}-x^{n-2}y+x^{n-3}y^2-x^{n-4}y^3+...-xy^{n-2}+y^{n-1})[/tex]

    However, if n is even, then [tex]x^n+y^n\neq 0[/tex] except for in the trivial case of x,y=0. This means you can't factor it over the reals. You'll need to use complex numbers. You could convert it into a few different ways, such as [tex]x^n-i^2y^n[/tex] and take difference of two squares, or, if you want to follow the same factorizing process as above, take [tex]x^n+(iy)^n[/tex] and take two cases, when [itexi^n[/itex] is equal to 1, and when equal to -1.
     
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