Difference between revisions of "1984 AIME Problems/Problem 8"

Revision as of 09:09, 14 March 2010

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 $\boxed{\theta=160}$ .

Also,

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

@@ Line 9: / Line 9: @@
 Also,
-From above, you notice that <math>z^6+z^3+1 = r^9-1/r^3-1</math>. 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 <math>\boxed{\theta=160}</math>.
+From above, you notice that <math>z^6+z^3+1 = </math>\frac {r^9-1}{r^3-1<math>}</math><math>. 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 </math>\boxed{\theta=160}$.
 == See also ==

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

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Difference between revisions of "1984 AIME Problems/Problem 8"

Revision as of 09:09, 14 March 2010

Problem

Solution

See also