Specialist2025 Exam 1 · Q8Tech-free5 marks

Conjugate roots and factorising a quartic over C

Consider the function with rule $f(z) = z^4 + 6z^2 + 25$ , where $z \in C$ .

Consider $z_1 = 1 + 2i$ .

Plot and label $z_1$ and $\overline{z}_1$ on the Argand plane below.

Blank Argand plane grid with Re(z) and Im(z) axes, each marked from −4 to 4 in steps of 2, with origin O, for plotting z1 and its conjugate.

1 mark

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

Given that $1 + 2i$ is a solution of $f(z) = 0$ , find a quadratic factor of $f(z)$ .

2 marks

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

Hence, find all remaining solutions of $f(z) = 0$ .

2 marks

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