Quartic roots and an Argand triangle · source reference

Question

A, B and C represent complex numbers $a=0,b=3,c=-2+i$. These numbers are roots of $f(z)=0$, where $f(z)$ is a quartic polynomial with real coefficients and $z$ is complex.

(a) Express $f(z)$ as a product of two quadratic factors with real coefficients. [3]

(b) Sketch an Argand diagram showing all roots. [2]

(c) W represents $w$, with $c=iw$. Find $\overrightarrow{AC}\cdot\overrightarrow{AW}$ and the area of triangle APW, where P represents $c-b$. [4]

[2024/DHS/Prelim/I/7]

Source working

Printed working

(a) Conjugate root theorem gives $-2-i$ as the fourth root.

$$\begin{aligned}f(z)&=z(z-3)(z-(-2+i))(z-(-2-i))\\&=(z^2-3z)((z+2)-i)((z+2)+i)\\&=(z^2-3z)((z+2)^2-i^2)\\&=(z^2-3z)(z^2+4z+5).\end{aligned}$$

(b) Argand diagram: $A=(0,0)$, $B=(3,0)$, $C=(-2,1)$, $D=(-2,-1)$, Re and Im axes, with ticks $-2,3,1,-1$.

(c) $\overrightarrow{AC}\cdot\overrightarrow{AW}=0$. W is C rotated $90^\circ$ clockwise about the origin; thus $\angle CAW=90^\circ$.

$$\overrightarrow{AW}=\begin{pmatrix}1\\2\\0\end{pmatrix},\qquad\overrightarrow{AP}=\begin{pmatrix}-5\\1\\0\end{pmatrix}.$$ $$\begin{aligned}\text{Area}(APW)&=\frac12|\overrightarrow{AP}\times\overrightarrow{AW}|\\&=\frac12\left|\begin{pmatrix}-5\\1\\0\end{pmatrix}\times\begin{pmatrix}1\\2\\0\end{pmatrix}\right|\\&=\frac12\left|\begin{pmatrix}0\\0\\-11\end{pmatrix}\right|=5.5\text{ units}^2.\end{aligned}$$

Qualification of the printed factorization

The question does not specify a monic polynomial. In general $f(z)=K(z^2-3z)(z^2+4z+5)$ for an arbitrary nonzero real $K$. The printed answer assumes $K=1$. Its roots and geometry are unaffected.

Printed answer key

(a) $(z^2-3z)(z^2+4z+5)$. (c) $0$, area $5.5$ units squared.

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 3; solution pages 17. These are original document page numbers, retained as text metadata.