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**Fratricide** Q1:

a) Find the exact solutions in C for the equation z^{2}-2√3z + 4 = 0, writing your solutions in cartesian form.

b) i) Plot the two solutions from part a on an Argand diagram.

ii) Find the equation of the circle, with centre the origin, which passes through these two points.

iii) Find the value of a ∈ Z such that the circle passes through (0, ± a).

iv) Let Q(z) = (z^{2} + 4)(z^{2}-2√3z + 4). Find the polynomial P(z) such that Q(z)P(z) = z^{6} + 64 and explain the significance of the result.

Parts a and b i, ii and iii were straight forward, but I've hit a wall with iv. z^{6} + 64 looks familiar, but I can't quite put my finger on it.