Finding specific root values

5 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + q z + 37 - 36 \mathrm { i } = 0$$ where \(p\) and \(q\) are constants, has three complex roots, \(\alpha , \beta\) and \(\gamma\). It is given that \(\beta = - 2 + 3 \mathrm { i }\) and \(\gamma = 1 + 2 \mathrm { i }\).

1. 1. Write down the value of \(\alpha \beta \gamma\).
   2. Hence show that \(( 8 + \mathrm { i } ) \alpha = 37 - 36 \mathrm { i }\).
   3. Hence find \(\alpha\), giving your answer in the form \(m + n \mathrm { i }\), where \(m\) and \(n\) are integers.
2. Find the value of \(p\).
3. Find the value of the complex number \(q\).

A cubic equation has roots \(\alpha\), \(\beta\), \(\gamma\) such that $$\alpha + \beta + \gamma = -9, \quad \alpha\beta + \beta\gamma + \gamma\alpha = 20, \quad \alpha\beta\gamma = 0.$$

1. Find the values of \(\alpha\), \(\beta\) and \(\gamma\). [4]
2. Find the cubic equation with roots \(3\alpha\), \(3\beta\), \(3\gamma\). Give your answer in the form \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), \(d\) are constants to be determined. [4]

The equation \(4x^4 - 4x^3 + px^2 + qx - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha\), \(-\alpha\), \(\beta\) and \(\frac{1}{\beta}\).

1. Determine the exact roots of the equation. [5]
2. Determine the values of \(p\) and \(q\). [4]

$$f(z) = z^3 - 8z^2 + pz - 24$$ where \(p\) is a real constant. Given that the equation \(f(z) = 0\) has distinct roots $$\alpha, \beta \text{ and } \left(\alpha + \frac{12}{\alpha} - \beta\right)$$

1. solve completely the equation \(f(z) = 0\) [6]
2. Hence find the value of \(p\). [2]