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Complex Numbers ~ Factors

  1. Mar 18, 2009 #1


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    1. The problem statement, all variables and given/known data
    Resolve [itex]z^5-1[/itex] into real linear and quadratic factors.

    Hence prove that [tex]cos\frac{2\pi}{5}+cos\frac{4\pi}{5}=-\frac{1}{2}[/tex]

    2. Relevant equations



    3. The attempt at a solution
    I was able to show that the the roots of [itex]z^5-1=0[/itex] are:


    And hence, the real factors are:


    But now I'm stuck and not sure how to start proving that last equation.
  2. jcsd
  3. Mar 18, 2009 #2


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    If you multiply your factored form back out again, then the identity you are trying to prove is the coefficient of z^4. It's also basically the sum of all of the five roots.
    Last edited: Mar 18, 2009
  4. Mar 18, 2009 #3


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    Aha and the coefficient of z4 is 0, so:


    Therefore, [tex]1+2cos\frac{2\pi}{5}+2cos\frac{4\pi}{5}=0[/tex]


    Thanks :smile:
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