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Math Help - need help with complex number

  1. #1
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    need help with complex number

    Hi, the question I'm struggle with is (z^2 + 1)^3 = -1
    I have to find all the solutions for this question and the answer should be in Cartesian form

    I start the question by powering both side by 1/3
    thus the equation become

    Z^2 + 1 = -1^(1/3)
    since i^2 = -1
    Z^2 +1 = (i^2)^(1/3)
    from here I sub in Z = X + iY
    so the equation become
    X^2 + 2XYi - Y^2 = (i^2)^(1/3) + 1

    I'm pretty much stuck now and unable to continue the question. Can you guy tell me how to continue this question and get the solution in cartesian form?
    (I have a feeling that I did something wrong in the step above
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: need help with complex number

    I think what I would do is write the equation as the sum of two cubes:

    (z^2+1)^3+1^3=0

    Factor:

    ((z^2+1)+1)((z^2+1)^2-(z^2+1)+1)=0

    (z^2+2)(z^4+z^2+1)=0

    (z^2+2)(z^2+z+1)(z^2-z+1)=0

    Now you can easily get all six roots in rectangular form.
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  3. #3
    Junior Member
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    Re: need help with complex number

    I have another tries with the question, thank to your reply I got an idea to move 1 to the other side first

    I started by +1 to both side of the equation then powered both side by (1/3)

    the equation become Z2 +2 = 0
    I then minus -2 to both side
    then sub z=x+iY
    (x+iY2) = -2
    X2 + 2XYi -Y2= -2
    (X2 + Y2) - 2XYi = -2
    I now have the answer in Cartesian form, however how do I find all the solutions?

    thank
    Last edited by junkwisch; May 12th 2013 at 12:27 AM.
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