The complex number $w$ is given by $1 + 2\mathrm{i}$. Find $w^3$ in the form \(x + \mathrm{i}y

Question

The complex number $w$ is given by $1 + 2\mathrm{i}$. Find $w^3$ in the form $x + \mathrm{i}y$, showing your working.[2] Given that $w$ is a root of the equation $az^3 + 5z^2 + 17z + b = 0$, find the values of the real numbers $a$ and $b$.[3] Using these values of $a$ and $b$, find all the roots of this equation in exact form.[3]

Accepted Answer

(i) $w^3=-11-2\mathrm{i}$ (ii) $a=27$, $b=295$ (iii) $\displaystyle z=1\pm2\mathrm{i},\ -\frac{59}{27}$

Video Solution:

Solution 1