2022 RI P2 Q2

Do not use a calculator in answering this question.

1. Let \(\mathrm{f}\left( z \right)\) be a polynomial in \(z\) of degree \(4\) with real coefficients. The equation \(\mathrm{f}\left( z \right)=0\) has four roots, namely \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) such that they satisfy the following two conditions:
   \(\alpha \,\beta \,\gamma \,\delta <0\) and \({{\alpha }^{2}}+{{\beta }^{2}}+{{\gamma }^{2}}+{{\delta }^{2}}<0\).
   Based on the two conditions, a student concludes that the equation \(\mathrm{f}\left( z \right)=0\) has one positive real root, one negative real root and a pair of complex conjugate roots.
   State, with reasons, whether the student’s claim is true.[3]
2. It is given that \(\mathrm{g}\left( z \right)={{z}^{4}}+{{z}^{3}}-2{{z}^{2}}+4z-24\).
   Verify that \(z=2\mathbf{i}\) is a root of the equation \(\mathrm{g}\left( z \right)=0\). Hence find the other roots of the equation.[5]