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Complex cube root of 1

  1. Feb 5, 2009 #1
    1. The problem statement, all variables and given/known data

    If w is a complex cube root of 1, prove that x + wy + w^2z is a factor of x^3+ y^3 + z^3 - 3xyz, and hence factorise the equation completely.

    2. Relevant equations

    Complex cube root of 1 = -1/2 +/- 3^1/2/2 i

    3. The attempt at a solution

    Erm, I feel way over my head. I have tried plugging in the equation to the first one but this doesn't seem to generate anything intelligible for me.
     
  2. jcsd
  3. Feb 5, 2009 #2

    gabbagabbahey

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    Hi Ferrus, welcome to PF:smile:

    Hint 1: If [itex]x + wy + w^2z[/itex] is a factor of [itex]x^3+ y^3 + z^3 - 3xyz[/itex], what can you say about the remainder of [tex]\frac{x^3+ y^3 + z^3 - 3xyz}{x + wy + w^2z}[/tex]

    Hint 2: If [itex]z^3=1[/itex], then [itex]z^3-1=(z-1)(z^2+z+1)=0[/itex].... so what can you say about [itex]w^2+w+1[/itex]?
     
  4. Feb 5, 2009 #3

    tiny-tim

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    Welcome to PF!

    Hi Ferrus ! Welcome to PF! :smile:

    Hint: if (x+a) is a factor of a polynomial, then put x = -a and the polynomial will be zero …

    so put x = … ? :wink:
     
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