8 The function \(\mathrm { f } ( z ) = z ^ { 4 } - z ^ { 3 } + a z ^ { 2 } + b z + c\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha , \beta\), \(\gamma\) and \(\delta\) where \(\alpha = 1\) and \(\beta = 1 + \mathrm { j }\).
- Write down the other complex root and explain why the equation must have a second real root.
- Write down the value of \(\alpha + \beta + \gamma + \delta\) and find the second real root.
- Find the values of \(a , b\) and \(c\).
- Write down \(\mathrm { f } ( - z )\) and the roots of \(\mathrm { f } ( - z ) = 0\).