N2024 P1 Q3

The equation \(z^3 + az^2 + bz + c = 0\), where \(a\), \(b\) and \(c\) are constants, has roots \(3\) and \(w\) where \(w\) is a complex number.

1. State a condition on \(a\), \(b\) and \(c\) for the third root to be \(w^*\).[1]

2. Given that the condition in part (a) holds, and that \(w = -1 + 2\mathrm{i}\), find the values of \(a\), \(b\) and \(c\).

   [3]