Let $P(z) = z^3 − 6z^2 + (21 + 4i)z − 80 − ki$. If $P(1 − 4i) = 0$, find the remaining two roots and the value of $k$.
What crossed my mind first was using Vieta's formulas and the given root. However, setting up the sum, product, and sum of products of two roots taken at a time produced:
$5+4i=(a+c)+(b+d)i$
$21+4i=(a+c+ac+4b+4d-bd)+(b+d+ad+bc-4a-4c)i$
$80+ki=(ac-bd+4ad+4bc)+(ad+bc-4ac+4bd)i$
Some of these expressions are rather unwieldy and I am not sure how to manipulate them. Does anyone have any suggestions?
Hint: by polynomial Euclidean division:
$$ z^3 - 6 z^2 + (21 + 4 i) z - 80 = \big(z^2 - (5 + 4 i) z + 20 i\big) \cdot \big(z - (1 - 4 i)\big) + 20 i $$