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Math Help - Complex number question

  1. #1
    Senior Member Stroodle's Avatar
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    Complex number question

    Hi, I'm having trouble solving part c of the following question.


    Let w=1+ai where a is a real constant.


    a. Show that \left | w^3 \right | =(1+a^2)^{\frac{3}{2}}

    This part I'm fine with as |w^3|=|w|^3


    b. Find the values of a for which |w^3|=8

    I get a=\pm\sqrt{3} for this part.


    c. Let p(z)=z^3+bz^2+cz+d where b,\ c and d are non-zero real constants. If  p(z)=0 for z=w and all roots of p(z)=0 satisfy |w^3|=8, find the values of b,\ c and d and show that these are the only possible values.

    Stuck on this part of the question...


    Thanks for your help!
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  2. #2
    Pim
    Pim is offline
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    p(z) has 3 roots. If they all satisfy |w^3|=8 It means they are all equal to 1+\sqrt{3} or 1-\sqrt{3}

    A third degree polynomial of this form can be constructed by the following method.

    p(z) = (z-z_1)(z-z_2)(z-z_3)

    Where z_1 z_2 z_3 are the roots of the polynomial.

    Is this enough to get you started?
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