2022 RI P2 Q2
Do not use a calculator in answering this question.
(a)
Let $\text{f}\left( z \right)$ be a polynomial in $z$ of degree $4$ with real coefficients. The equation $\text{f}\left( z \right)=0$ 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 $\text{f}\left( z \right)=0$ 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]
(b)
It is given that $\text{g}\left( z \right)={{z}^{4}}+{{z}^{3}}-2{{z}^{2}}+4z-24$.
Verify that $z=2\mathbf{i}$ is a root of the equation $\text{g}\left( z \right)=0$. Hence find the other roots of the equation.
[5]
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