1984 AIME Problems/Problem 8

Revision as of 09:09, 14 March 2010 by Aplus95 (talk | contribs) (Solution)


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 $\theta$.


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 $\boxed{\theta=160}$.


From above, you notice that $z^6+z^3+1 = \frac {r^9-1}{r^3-1$ (Error compiling LaTeX. ! File ended while scanning use of \frac .)}$. Therefore, the solutions are all of the ninth roots of unity that are not the third roots of unity. After checking, the only angle is$\boxed{\theta=160}$.

See also

1984 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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All AIME Problems and Solutions
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