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Home » Square's perimeter: \(4 \times 10 = 40\) meters. - Sterling Industries

Square's perimeter: \(4 \times 10 = 40\) meters. - Sterling Industries

📅 April 17, 2026 👤 scraface

Apr 17, 2026

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📌 Question: Let $ P(x) $ be a cubic polynomial such that $ P(1) = 1 $, $ P(2) = 4 $, $ P(3) = 9 $, and $ P(4) = 16 $. Find $ P(5) $.

📌 We are given that $ P(x) $ is a cubic polynomial satisfying:

📌 Then $ Q(x) $ is also a polynomial, and:

📌 So define $ P(x) $, cubic, satisfying:

📌 Use Lagrange interpolation for $ P(x) $, cubic, through $ (1,1), (2,4), (3,9), (4,16) $.

📌 Let $ P(x) = a(x-2)(x-3)(x-4) + b(x-1)(x-3)(x-4) + c(x-1)(x-2)(x-4) + d(x-1)(x-2)(x-3) $.

📌 $ x = 3 $: $ 27a + 9b + 3c + d = 9 $

📌 $ x = 4 $: $ 64a + 16b + 4c + d = 16 $

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