Specialist2024 Exam 1 · Q1Tech-free4 marks

Factorising a complex cubic and plotting its roots

Consider the function with rule $f(z) = 3z^3 + 2iz^2 + 3z + 2i$ , where .

Verify that $3z + 2i$ is a factor of $f(z)$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Hence or otherwise, solve the equation $f(z) = 0$ .

Give your answers in Cartesian form.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Plot the solutions of $f(z) = 0$ on the Argand diagram below.

Blank Argand diagram with Re(z) and Im(z) axes marked from −2 to 2, overlaid with a polar grid of concentric circles at radii that are multiples of 1/3 and radial lines every 30 degrees.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.