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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?

equation

Solution I

Multiply the equation by z3:

equation

and take advantage of the fact that points on the unit circle have absolute value 1:

equation

The equation has now become a cyclotomic equation:

equation

equation

Since:

equation

We obtain for z:

equation

equation

equation

equation

equation

equation

equation

The solution is formed by the 6 points in the figure:

figure

Solution II

Factor the equation:

equation

Write it as sums and products of conjugate numbers:

equation

Now take advantage of:

equation

equation

and use the properties of conjugates:

equation

equation

The equation becomes:

equation

with solutions when:

equation

or

equation

Keeping in mind that z is restricted to the unit circle, we obtain the same 6 points as in the previous solution approach.