Apply the factor theorem and the remainder theorem for polynomials.

Given \(P(z) = z^3 + (2 - i)z^2 + az + b\), where \(a, b \in \mathbb{C}\), the polynomial has \((z - (1 + i))\) as a factor. Additionally, when \(P(z)\) is divided by \((z + 2)\), the remainder is \(6 + 2i\). (a) Use the factor theorem to find the value of \(a\). (2 marks) (b) Use the remainder theorem to find the value of \(b\). (2 marks)

Given that \(P(z) = z^3 + (2-i)z^2 - 4z + k\), where \(k \in \mathbb{C}\), has a remainder of \(5 + 3i\) when divided by \(z + 1\), apply the remainder theorem to determine the value of \(k\).

Given that \( P(z) = z^3 + (2 - i)z^2 + az + b \) and \( Q(z) = z^2 - 3iz + c \), where \( a, b, c \in \mathbb{C} \), have the same remainder when divided by \( z + i \), and \( (z - 2i) \) is a factor of \( P(z) \), determine the values of \( a \) and \( b \).

The polynomial \(P(z) = z^3 + 2iz^2 + kz + 9i\), where \(k \in \mathbb{C}\), has a factor of \((z + 3i)\). Use the factor theorem to determine the value of \(k\).