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Thread: 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 $\displaystyle w=1+ai$ where a is a real constant.


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

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


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

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


    c. Let $\displaystyle p(z)=z^3+bz^2+cz+d$ where $\displaystyle b,\ c $ and $\displaystyle d$ are non-zero real constants. If $\displaystyle p(z)=0$ for $\displaystyle z=w$ and all roots of $\displaystyle p(z)=0$ satisfy $\displaystyle |w^3|=8$, find the values of $\displaystyle b,\ c$ and $\displaystyle d$ 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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    $\displaystyle p(z)$ has 3 roots. If they all satisfy $\displaystyle |w^3|=8$ It means they are all equal to $\displaystyle 1+\sqrt{3}$ or $\displaystyle 1-\sqrt{3}$

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

    $\displaystyle p(z) = (z-z_1)(z-z_2)(z-z_3)$

    Where $\displaystyle z_1 z_2 z_3$ are the roots of the polynomial.

    Is this enough to get you started?
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