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Math Help - Complex Numbers: Roots of an EQN

  1. #1
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    Complex Numbers: Roots of an EQN

    Hi guys, need help with this question:

    Show that the roots of the EQN z^5 - (z - i)^5 = 0, z is not equal to i, are
    (1/2)[cot(k(pi)/5) + i] where k = 1,2,3,4.

    Thanks in advance!
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  2. #2
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    Since z = i is not a solution, I'm puzzled by the question.
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  3. #3
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    Quote Originally Posted by Blizzardy View Post
    Hi guys, need help with this question:

    Show that the roots of the EQN z^5 - (z - i)^5 = 0, z is not equal to i, are
    (1/2)[cot(k(pi)/5) + i] where k = 1,2,3,4.

    Thanks in advance!
    Why not start by expanding \displaystyle (z- i)^5 and simplifying before trying to solve the equation?
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  4. #4
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    Try this line of reasoning:

    z^{5}-(z-i)^{5}=0

    z^{5}=(z-i)^{5}

    \left(\frac{z-i}{z}\right)^{\!\!5}=1.

    Let

    w=\frac{z-i}{z}.

    Then w^{5}=1, the fifth roots of unity (exactly). So I would try writing down the w's, and then solving zw=z-i for z once you've done that.

    Make sense?
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