The equation z^6+z^3+1=0 has one complex root with argument θ between 90° and 180° in the complex plane. Determine the degree measure of θ.

1984 AIME Problem 8

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The equation

z^6+z^3+1=0

has one complex root with argument

\theta

between

90^\circ

and

180^\circ

in the complex plane. Determine the degree measure of

\theta.

Answer

Substitute

u=z^3

The two possible arguments of

z^3

are

120^\circ

and

240^\circ

Let

u=z^3.

Then

u^2+u+1=0,

u

has argument

120^\circ

240^\circ.

Taking cube roots gives possible arguments

40^\circ,160^\circ,280^\circ,80^\circ,200^\circ,320^\circ.

The only one strictly between

90^\circ

and

180^\circ

160^\circ.