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

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The equation <math>z^6+z^3+1</math> has [[complex root]]s with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in the [[complex plane]]. Determine the degree measure of <math>\theta</math>.
 
The equation <math>z^6+z^3+1</math> has [[complex root]]s with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in the [[complex plane]]. Determine the degree measure of <math>\theta</math>.
  
== Solution ==
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== Solution 1 ==
 
If <math>r</math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math>. The polynomial <math>x^9-1</math> has all of its roots with [[absolute value]] <math>1</math> and argument of the form <math>40m^\circ</math> for integer <math>m</math>.
 
If <math>r</math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math>. The polynomial <math>x^9-1</math> has all of its roots with [[absolute value]] <math>1</math> and argument of the form <math>40m^\circ</math> for integer <math>m</math>.
  
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From above, you notice that <math>z^6+z^3+1 = \frac {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 = \frac {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>.
  
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== Solution 2 ==
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Note that the substitution <math>y=z^3</math> simplifies this to <math>y^2+y+1</math>. Simply applying the quadratic formula gives roots <math>y_{1,2}=\frac{1}{2}\pm \frac{\sqrt{3}i}{2}</math>, which have angles of 120 and 240, respectively. This means <math>arg(z) = \frac{120,240}{3} + \frac{360n}{3}</math>, and the only one between 90 and 180 is <math>\boxed{\theta=160}</math>.
 
== See also ==
 
== See also ==
 
{{AIME box|year=1984|num-b=7|num-a=9}}
 
{{AIME box|year=1984|num-b=7|num-a=9}}
  
 
[[Category:Intermediate Trigonometry Problems]]
 
[[Category:Intermediate Trigonometry Problems]]

Revision as of 23:41, 10 February 2011

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 1

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 $z^6+z^3+1 = \frac {r^9-1}{r^3-1}$. 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 $\boxed{\theta=160}$.

Solution 2

Note that the substitution $y=z^3$ simplifies this to $y^2+y+1$. Simply applying the quadratic formula gives roots $y_{1,2}=\frac{1}{2}\pm \frac{\sqrt{3}i}{2}$, which have angles of 120 and 240, respectively. This means $arg(z) = \frac{120,240}{3} + \frac{360n}{3}$, and the only one between 90 and 180 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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