a) Find the exact solutions in C for the equation z2-2√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) = (z2 + 4)(z2-2√3z + 4). Find the polynomial P(z) such that Q(z)P(z) = z6 + 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. z6 + 64 looks familiar, but I can't quite put my finger on it.