Determine and use linear factors of quadratic polynomials with real coefficients that involve the complex conjugate root theorem, e.g. determine the coefficients of a real quadratic equation given one complex root.

An electrical engineer is analysing the impedance of a circuit component modelled by a quadratic equation \(Z^2 + pZ + q = 0\), where \(p, q \in \mathbb{R}\) and \(Z\) represents the complex impedance in ohms. One of the impedance values is \(Z = 3 - 4i\) ohms. (a) Determine the value of \(p\) and \(q\). (2 marks) (b) Hence express the quadratic in factorised form. (1 mark)

A quadratic equation with real coefficients has \(z = 2 + 3i\) as one of its roots. What is the product of the roots?

A quadratic polynomial \(Q(z) = z^2 + mz + n\), where \(m, n \in \mathbb{R}\), has \(z = 3 - 4i\) as one of its roots. (a) State the second root of \(Q(z)\). (1 mark) (b) Determine the values of \(m\) and \(n\). (2 marks)