Difference between revisions of "1996 AIME Problems/Problem 11"

Revision as of 19:17, 27 November 2007

Problem

Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$ , where $0<r$ and $0\leq \theta <360$ . Find $\theta$ .

Solution

Solution 1

$\begin{eqnarray*} 0 &=& z^6 - z + z^4 + z^3 + z^2 + z + 1 = z(z^5 - 1) + \frac{z^5-1}{z-1}\\ 0 &=& \frac{(z^5 - 1)(z(z-1)+1)}{z-1} = \frac{(z^2-z+1)(z^5-1)}{z-1} \end{eqnarray*}$

Thus $z^5 = 1, z \neq 1 \Longrightarrow z = \mathrm{cis}\ 72, 144, 216, 288$ , or $z^2 - z + 1 = 0 \Longrightarrow z = \frac{1 \pm \sqrt{-3}}{2} = \mathrm{cis}\ 60, 300$ (see cis). Discarding the roots with negative imaginary parts (leaving us with $\cis \theta, 0 < \theta < 180$ (Error compiling LaTeX. Unknown error_msg)), we are left with $\mathrm{cis}\ 60, 72, 144$ ; their product is $P = \mathrm{cis} 60 + 72 + 144 = \mathrm{cis} \boxed{276}$ .

Solution 2

Let $w =$ the fifth roots of unity, except for $1$ . Then $w^6 + w^4 + w^3 + w^2 + 1 = w^4 + w^3 + w^2 + w + 1 = 0$ , and since both sides have the fifth roots of unity as roots, we have $z^4 + z^3 + z^2 + z + 1 | z^6 + z^4 + z^3 + z^2 + 1$ . Long division quickly gives the other factor to be $z^2 - z + 1$ . The solution follows as above.

Solution 3

Divide through by $z^3$ . We get the equation $z^3 + \frac {1}{z^3} + z + \frac {1}{z} + 1 = 0$ . Let $x = z + \frac {1}{z}$ . Then $z^3 + \frac {1}{z^3} = x^3 - 3x$ . Our equation is then $x^3 - 3x + x + 1 = x^3 - 2x + 1 = (x - 1)(x^2 + x - 1) = 0$ , with solutions $x = 1, \frac { - 1\pm\sqrt {5}}{2}$ . For $x = 1$ , we get $z = \text{cis}60,\text{cis}300$ . For $x = \frac { - 1 + \sqrt {5}}{2}$ , we get $z = \text{cis}{72},\text{cis}{292}$ (using exponential form of $\cos$ ). For $x = \frac { - 1 - \sqrt {5}}{2}$ , we get $z = \text{cis}144,\text{cis}216$ . The ones with positive imaginary parts are ones where $0\le\theta\le180$ , so we have $60 + 72 + 144 = 276$ .

@@ Line 1: / Line 1: @@
 == Problem ==
-Let <math>\mathrm {P}</math> be the product of the roots of <math>z^6+z^4+z^3+z^2+1=0</math> that have an imaginary part, and suppose that <math>\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</math> and <math>0\leq \theta <360</math>. Find <math>\theta</math>.
+Let <math>\mathrm {P}</math> be the product of the [[root]]s of <math>z^6+z^4+z^3+z^2+1=0</math> that have a positive [[imaginary]] part, and suppose that <math>\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</math> and <math>0\leq \theta <360</math>. Find <math>\theta</math>.
 __TOC__
@@ Line 7: / Line 7: @@
 <cmath>\begin{eqnarray*}
 &=& z^6 - z + z^4 + z^3 + z^2 + z + 1 = z(z^5 - 1) + \frac{z^5-1}{z-1}\\
-&=& \frac{(z^5 - 1)(z(z-1)+1)}{z-1}\\
+&=& \frac{(z^5 - 1)(z(z-1)+1)}{z-1} = \frac{(z^2-z+1)(z^5-1)}{z-1}
-&=& \frac{(z^2-z+1)(z^5-1)}{z-1}
 \end{eqnarray*}</cmath>
-Thus <math>z^5 = 1, z \neq 1 \Longrightarrow z = \mathrm{cis} \frac{360 k}{5}, k = 1, 2, \ldots 4</math>, or <math>z^2 - z + 1 = 0 \Longrightarrow z = \frac{1 \pm \sqrt{-3}}{2} = \mathrm{cis} 60, 300</math>. Therefore all of the roots are complex, and the answer is <math>\boxed{0}</math>.
+Thus <math>z^5 = 1, z \neq 1 \Longrightarrow z = \mathrm{cis}\ 72, 144, 216, 288</math>, or <math>z^2 - z + 1 = 0 \Longrightarrow z = \frac{1 \pm \sqrt{-3}}{2} = \mathrm{cis}\ 60, 300</math> (see [[cis]]). Discarding the roots with negative imaginary parts (leaving us with <math>\cis \theta, 0 < \theta < 180</math>), we are left with <math>\mathrm{cis}\ 60, 72, 144</math>; their product is <math>P = \mathrm{cis} 60 + 72 + 144 = \mathrm{cis} \boxed{276}</math>.
 === Solution 2 ===
+Let <math>w = </math> the fifth [[roots of unity]], except for <math>1</math>. Then <math>w^6 + w^4 + w^3 + w^2 + 1 = w^4 + w^3 + w^2 + w + 1 = 0</math>, and since both sides have the fifth roots of unity as roots, we have <math>z^4 + z^3 + z^2 + z + 1  | z^6 + z^4 + z^3 + z^2 + 1</math>. Long division quickly gives the other factor to be <math>z^2 - z + 1</math>. The solution follows as above.
+=== Solution 3 ===
+Divide through by <math>z^3</math>. We get the equation <math>z^3 + \frac {1}{z^3} + z + \frac {1}{z} + 1 = 0</math>. Let <math>x = z + \frac {1}{z}</math>. Then <math>z^3 + \frac {1}{z^3} = x^3 - 3x</math>. Our equation is then <math>x^3 - 3x + x + 1 = x^3 - 2x + 1 = (x - 1)(x^2 + x - 1) = 0</math>, with solutions <math>x = 1, \frac { - 1\pm\sqrt {5}}{2}</math>. For <math>x = 1</math>, we get <math>z = \text{cis}60,\text{cis}300</math>. For <math>x = \frac { - 1 + \sqrt {5}}{2}</math>, we get <math>z = \text{cis}{72},\text{cis}{292}</math> (using exponential form of <math>\cos</math>). For <math>x = \frac { - 1 - \sqrt {5}}{2}</math>, we get <math>z = \text{cis}144,\text{cis}216</math>. The ones with positive imaginary parts are ones where <math>0\le\theta\le180</math>, so we have <math>60 + 72 + 144 = 276</math>.
 == See also ==

1996 AIME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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