8 Stephen is correctly told that \(( 1 + \mathrm { i } )\) and - 1 are two roots of the polynomial equation
$$z ^ { 3 } - 2 \mathrm { i } z ^ { 2 } + p z + q = 0$$
where \(p\) and \(q\) are complex numbers.
8
- Stephen states that ( \(1 - \mathrm { i }\) ) must also be a root of the equation because roots of polynomial equations occur in conjugate pairs.
Explain why Stephen's reasoning is wrong.
8
- \(\quad\) Find \(p\) and \(q\)