8 A polynomial \(\mathrm { P } ( z )\) has real coefficients. Two of the roots of \(\mathrm { P } ( z ) = 0\) are \(2 - \mathrm { j }\) and \(- 1 + 2 \mathrm { j }\).
- Explain why \(\mathrm { P } ( z )\) cannot be a cubic.
You are given that \(\mathrm { P } ( z )\) is a quartic.
- Write down the other roots of \(\mathrm { P } ( z ) = 0\) and hence find \(\mathrm { P } ( z )\) in the form \(z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d\).
- Show the roots of \(\mathrm { P } ( z ) = 0\) on an Argand diagram and give, in terms of \(z\), the equation of the circle they lie on.