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Homework Help: Complex Numbers: 4th Degree Polynomial

  1. Aug 7, 2010 #1
    1. The problem statement, all variables and given/known data

    Solve the following equation:

    [tex]z^4+z^3+z^2+z+1 = 0[/tex]

    z is a complex number.

    2. The attempt at a solution
    I was trying to factorize it to 1st degree polynomial multiplied by 3rd degree polynomial:
    [tex](z+a)(z^3+bz^2+cz+1/a) = 0[/tex]
    I discovered that I need to solve 3rd degree polynomial just to do that.
    [tex]a^3-a^2+1 = 0[/tex]
    This is too much mess for a small homework exercise. I think that there is a technique that I am not aware of.




    Thank You.
     
  2. jcsd
  3. Aug 7, 2010 #2
    Try multiplying the original equation by z - 1.
     
  4. Aug 7, 2010 #3

    Hurkyl

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    The lesson here is that you shouldn't forget what you learned in earlier classes. That polynomial is a geometric series, is it not?
     
  5. Aug 7, 2010 #4
    @Hurkyl Very perceptive of you ! :)

    Through the use of the finite geometric series sum you can find all the roots !

    @Petek it's the same as multiplying by (1-z), you right.
     
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