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Math Help - Root of complex polynomial

  1. #1
    JML
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    Root of complex polynomial

    I was asked if it is possible to solve this question with the complex exponential:

    (z + i)^5 - (z-i)^5=0

    I can see how this is solved by expanding out, but is there an easier or more systematic way to go about these problems?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well it is

    (z+i)^5=(z-i)^5

    Taking fifth roots on both sides, we have

    \omega(z+i)=z-i

    where \omega^5=1 is any fifth root of unity. Thus

    z(1-\omega)=i(1+\omega)

    or z=\frac{i(1+\omega)}{1-\omega}.
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