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quagmire the family guy - Appcentric

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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Solution: Consider the equation:
z^6 + z^3 + 1 = 0.
Let $ w = z^3 $. Then:
So $ z^3 = \omega $ or $ z^3 = \omega^2 $, where $ \omega = e^{2\pi i...
We now solve $ z^3 = \omega $ and $ z^3 = \omega^2 $. Each has three c...
First, write $ \omega = e^{2\pi i / 3} $, so the cube roots are:
z = \omega^{1/3} = e^{2\pi i / 9},\ e^{2\pi i (1+3)/9} = e^{8\pi i / 9...
Similarly, $ z^3 = \omega^2 = e^{-2\pi i / 3} $, so roots:
z = e^{-2\pi i / 9},\ e^{-8\pi i / 9},\ e^{-14\pi i / 9} = e^{4\pi i /...
So the six roots are:

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