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how to filter water in the wild - Redraw

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 14\mathbf{v} = \langle 8, 2, -4 \rangle \quad \Rightarrow \quad \mathbf{v} = \frac{1}{14} \langle 8, 2, -4 \rangle = \left\langle \frac{4}{7}, \frac{1}{7}, -\frac{2}{7} \right\rangle

📌 Thus, the vector is $ \boxed{\left\langle \dfrac{4}{7}, \dfrac{1}{7}, -\dfrac{2}{7} \right\rangle} $.

📌 Question: Among all roots of the polynomial $ z^6 - z^4 + z^2 - 1 = 0 $, a historian of mathematical astronomy seeks the maximum imaginary part, expressed as $ \sin \theta $ for $ 0 < \theta < \pi $. Find $ \theta $.

📌 Solution: Consider the polynomial:

📌 z^6 - z^4 + z^2 - 1 = 0

📌 So the roots satisfy either $ z^2 - 1 = 0 $ or $ z^4 + 1 = 0 $.

📌 Roots from $ z^2 - 1 = 0 $:

📌 $ z = \pm 1 $, both real, so imaginary part is 0.

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