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resc paragraph can't contain keyword, here are fresh options: - ECD Germany

📅 May 10, 2026 👤 scraface

May 10, 2026

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📌 x = rac{6 \pm 2\sqrt{3}}{6} = 1 \pm rac{\sqrt{3}}{3}

📌 Thus, the critical points are:

📌 oxed{1 + rac{\sqrt{3}}{3}, 1 - rac{\sqrt{3}}{3}}

📌 Question:** A quantum machine learning algorithm designer models the probability amplitude for a quantum state transition with the function \( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \). Verify if \( x = 1 \) is a root of multiplicity greater than 1.

📌 Solution:** To verify if \( x = 1 \) is a root of multiplicity greater than 1 for \( p(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \), we first check if \( p(1) = 0 \).

📌 Calculate \( p(1) \):

📌 p(1) = 1^4 - 4 imes 1^3 + 6 imes 1^2 - 4 imes 1 + 1 = 1 - 4 + 6 - 4 + 1 = 0

📌 Since \( p(1) = 0 \), \( x = 1 \) is indeed a root. To check the multiplicity, we compute the derivative \( p'(x) \):

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