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All the roots of the equation (1 + z)⁶ + z⁶ = 0:

1. lie on a unit circle centred at the origin
2. lie on a straight line in the complex plane
3. lie on the vertices of a regular polygon centred at the origin
4. have real part equal to -1/2

Correct answer: have real part equal to -1/2

Solution

((1+z)/z)⁶ = -1 forces |1+z| = |z|, the perpendicular bisector of 0 and -1, which is the vertical line Re(z) = -1/2.

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