1. ## 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...

2. $p(z)$ has 3 roots. If they all satisfy $|w^3|=8$ It means they are all equal to $1+\sqrt{3}$ or $1-\sqrt{3}$
$p(z) = (z-z_1)(z-z_2)(z-z_3)$
Where $z_1 z_2 z_3$ are the roots of the polynomial.