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📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Let us factor this expression. Group terms as follows:

📌 The root on the positive imaginary axis is:

📌 Note that all roots lie on or inside the unit circle because $P(z)$ is palindromic with positive coefficients, and no root lies exactly on the unit circle with non-zero real part unless its a root of unity, which is not the case here.

📌 Check $f(-1) = -1 -1 -1 -1 + 1 = -3$, $f(0) = 1$, $f(1) = 5$.

📌 There is exactly one real root in $(-1, 0)$, since $f(-1) < 0$, $f(0) > 0$.

📌 Therefore, the number of roots with positive imaginary part is:

📌 Question: A geographer is analyzing spatial patterns using trigonometric identities. Find the maximum value of $$

📌 We want to find the maximum value of this expression over all real $t

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