IAL 2025 May FP1 Q7

Question

[!problem]

$f(z) = Pz^4 - 36z^3 + Qz^2 + 192z + 68$

where $P$ and $Q$ are real constants.

Given that $3 + 5i$ is a root of the equation $f(z) = 0$

(a) write down another root of the equation $f(z) = 0$.

(1)

(b) Hence determine a quadratic factor, with real coefficients, of $f(z)$.

(2)

(c) Determine the value of $P$ and the value of $Q$.

(5)

Without using a calculator,

(d) determine the other roots of the equation $f(z) = 0$ giving your answers in simplest form.

(2)