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Complex polynomial- HELP

  1. Aug 23, 2009 #1
    1. The problem statement, all variables and given/known data

    (b) Using your answer to part (a), write down all complex roots of the polynomial
    z4 + iz3 - z2 - iz + 1:

    2. Relevant equations

    (a) Determine all of the complex solutions of
    z5 - i = 0:
    Write your answers in polar form with -pi < x < pi.

    3. The attempt at a solution

    I've worked out all of the complex solutions of part a to be :

    0= 1e^(ipi/10)

    1= 1e^(ipi/2)

    2= 1e^(9pi/10)

    3= 1e^ (-7pi/10)

    4= 1e^(-3pi/10)

    But I have no idea how to go about the second part, having only solved complex polynomials with Z not iZ. :S
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Aug 23, 2009 #2


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    Use the fact that [itex]z^5- a^5= (z- a)(z^4+ az^3+ a^2z^2+ a^3z+ a^4)[/itex]
    With a= i, using the facts that [itex]i^2= -1[/itex], [itex]i^3= -i[/itex], [itex]i^4= 1[/itex], and [itex]i^5= i[/itex], that becomes [itex]z^5- i= (z- i)(z^4+ iz^3- z^2- iz^3+ 1[/itex].

    Do you see the point?
  4. Aug 23, 2009 #3
    okay, I understand how that is the expanded form.
    So would the solutions for part b) just be the polar forms of part a) divided by (z-i)?

    Which would in turn make it:






  5. Aug 23, 2009 #4


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    Gold Member

    The answer to part (b) would be easier to see if you convert the roots of the polynomial from complex exponentials to complex numbers in the form z = a + i b. Once you've done this, is there another way you can express the equation z5 - i = 0 ?
  6. Aug 23, 2009 #5
    Are you by any chance taking calculus 2 at melbourne university? LoL
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