Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 1 / 11 / b

In polar form, a cube root of one has modulus one and argument $2\pi k/3$. The three roots of unity areThey are equally spaced on the unit circle.

Write $-8=8e^{i(\pi +2\pi k)}$. Its cube roots have modulus two and arguments $(\pi +2\pi k)/3$, for $k=0,1,2$. ThusEach cube equals $-8$, and a cubic polynomial has no further distinct roots.