N2013 P1 Q4

The complex number \(w\) is given by \(1 + 2\mathrm{i}\).

1. Find \(w^3\) in the form \(x + \mathrm{i}y\), showing your working.[2]
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]
3. Using these values of \(a\) and \(b\), find all the roots of this equation in exact form.[3]