Question 1403a

Complex Numbers

Conjugate Root Theorem

Question

It is given that

{z_1 = 1 + 4 \mathrm{i}}

is a root of the equation

z^3 - 5 z^2 + 23 z - 51=0.

The equation has two other roots,

{z_2}

and

{z_3,}

where

{z_2}

is complex and

{z_3}

is real. Find

{z_3.}

{z_3 = }