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This problem trains for: AMC-10, AMC-12, AIME.
If z is a complex number that can be represented as a point on the unit circle in the complex (Argand) plane, what is the representation of the solution of the following equation?
Multiply the equation by z3:
and take advantage of the fact that points on the unit circle have absolute value 1:
The equation has now become a cyclotomic equation:
We obtain for z:
The solution is formed by the 6 points in the figure:
Factor the equation:
Write it as sums and products of conjugate numbers:
Now take advantage of:
and use the properties of conjugates:
The equation becomes:
with solutions when:
Keeping in mind that z is restricted to the unit circle, we obtain the same 6 points as in the previous solution approach.