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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

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    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:

    (z5-a5)/(z1-a3)

    =

    z4-a2

    =

    z4-1

    ??
     
  5. Aug 23, 2009 #4

    kuruman

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    Homework Helper
    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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