2024 H2 Mathematics Paper 1 Question 3

Complex Numbers

Answers

$a, b$ and $c$ must be real.

$a = - 1,$ $b = - 1,$ $c = - 15.$

(a)

$a, b$ and $c$ must be real $∎$

(b)

Since all coefficients are real, the conjugate $w^{*} = - 1 - 2 i$ is the third root of the equation.

\begin{aligned} z^{3} + a z^{2} + b z + c \\ = (z - 3) (z - (- 1 + 2 i)) (z - (- 1 - 2 i)) \\ = (z - 3) (z + 1 - 2 i) (z + 1 + 2 i) \\ = (z - 3) ({(z + 1)}^{2} - 4 i^{2}) \\ = (z - 3) (z^{2} + 2 z + 5) \\ = z^{3} - z^{2} - z - 15 \end{aligned}

Comparing coefficients,

a = - 1, b = - 1, c = - 15 ∎