One of the roots of the equation $z^3+pz^2-z+q=0$, where $p$ and $q$ are real numbers, is

Question

One of the roots of the equation $z^3+pz^2-z+q=0$, where $p$ and $q$ are real numbers, is $1+2\mathrm{i}$. Find the other roots of the equation, the value of $p$ and the value of $q$.[5] The points $A$ and $B$ on the Argand diagram represent the complex numbers $w_1$ and $w_2$ respectively. Argand diagram as shown in the question The complex number $w_3$ is defined as $w_3=w_1-w_2$, and the complex number $w_4$ is defined as $w_4=w_2-w_1$. On the copy of the Argand diagram in the Printed Answer Booklet, mark the point $C$ that represents $w_3$ and the point $D$ that represents $w_4$. State the geometrical relationship between points $C$ and $D$.[3] The points $E$ and $F$ represent the complex numbers $w_1^*$ and $w_2^*$ respectively. Describe the transformation that maps the line $AB$ onto the line $EF$.[2]

Accepted Answer

(a) Other roots $1-2\mathrm{i}$ and $-3$; $p=1$, $q=15$ (b)(i) $C$ and $D$ are reflections of each other in the origin, so $O$ is the midpoint of $CD$ (ii) A reflection in the real axis

N2026 P2 Q3

N2026 P2 Q3