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Home » You Won’t Believe Which Justice League Character Has the HIDDEN Power! - Silent Sales Machine

You Won’t Believe Which Justice League Character Has the HIDDEN Power! - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 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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This value is achievable when $ x = y = z $, so the minimum is
oxed{3 - \log 2}.
Question: Find the remainder when $ (x^2 + 1)^{2025} + 1 $ is divided...
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x^6 - 1 = (x^2 - 1)(x^4 + x^2 + 1),
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