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1984 AIME Problems/Problem 8

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Problem

The equation z^6+z^3+1 has complex roots with argument \theta between 90^\circ and 180^\circ in the complex plane. Determine the degree measure of \theta.

Solution

If r is a root of z^6+z^3+1, then 0=(r^3-1)(r^6+r^3+1)=r^9-1. The polynomial x^9-1 has all of its roots with absolute value 1 and argument of the form 40m^\circ for integer m.

This reduces \theta to either 120^{\circ} or 160^{\circ}. But \theta can't be 120^{\circ} because if r=\cos 120^\circ +i\sin 120^\circ, then r^3=1 and r^6+r^3+1=3, a contradiction. This leaves \boxed{\theta=160}.

See also

1984 AIME (Problems • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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