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Home » The hypotenuse \(c\) of the right triangle can be found using the Pythagorean theorem: - AMAZONAWS

The hypotenuse \(c\) of the right triangle can be found using the Pythagorean theorem: - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 Let \( f(u) = u^5 - 3u^3 + 2u + 8 \). Since the divisor is quadratic, the remainder is linear: \( R(u) = au + b \).

📌 By the Remainder Theorem for polynomials, we use polynomial division or the method of undetermined coefficients. Alternatively, use substitution at the double root.

📌 \[ f(u) = (u - 1)^2 Q(u) + au + b \]

📌 \( f(1) = a(1) + b = a + b \)

📌 Differentiate: \( f'(u) = 2(u - 1)Q(u) + (u - 1)^2 Q'(u) + a \), so \( f'(1) = a \)

📌 Compute \( f(1) \):

📌 \[ f(1) = 1^5 - 3(1)^3 + 2(1) + 8 = 1 - 3 + 2 + 8 = 8 \]

📌 \[ a + b = 8 \quad ext{(1)} \]

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