2024 SJI P1 Q9

Consider the polynomial \(\mathrm p(x)=x^4+(k-4)x^3+(6-4k)x^2+(5k-4)x+5\), where \(x\in\mathbb C\) and \(k\in\mathbb R\). It is given that \(2+\mathrm i\) is a root of \(\mathrm p(x)=0\).

1. Write down another root.[1]
2. Determine the range of values of \(k\) for which \(\mathrm p(x)=0\) has two real roots.[6]

Suppose \(k=1\).

3. Find the zeros of \(\mathrm p(x)\).[3]
4. A polynomial, \(\mathrm q(w)\), has zeros that are reciprocals of the zeros of \(\mathrm p(x)\). Find an expression for \(\mathrm q(w)\).[3]