<s>[INST] Hi! Just wondering if any one can help me out with this roots question? (i). Factorise $z^2 - 5z + 6$ and hence, solve the equation $ z^2 - 5z + 6 = 0$
(ii). Show that $z^2 - 5z + 6$ is a factor of $z^3 + (-4 + i)z^2 + (1 - 5i)z + 6(1 + i)$.
(iii). Find the three roots of the equation $z^3 + (-4 + i)z^2 + (1 - 5i)z + 6(1 + i)$
I'm ok with the first two parts of the question i'm just a bit burned out on the third any help be much appreciated
 [/INST] (iii) :- $$z^3 + (-4 + i)z^2 + (1 - 5i)z + 6(1 + i)$$ 
$$= z^3 + -4z^2 + iz^2 + 1z - 5iz + 6 + 6i$$ 
$$= z^3  -4z^2 + z + 6 + i(z^2 - 5z + 6 )$$ 
By trial and error: $ z^3  -4z^2 + z + 6 = (z -2)(z^2 -2z - 3)$
$$(z -2)(z^2 -2z - 3) + i(z-3)(z -2)$$ 
$$= (z -2)(z^2 -2z - 3) + i(z-3)(z -2)$$ 
$$= (z -2)(z + 1)(z - 3) + i(z-3)(z -2)$$ 
$$= (z -2)(z -3)(z + 1 + i)$$ 
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