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HMMT 二月 2012 · 冲刺赛 · 第 10 题

HMMT February 2012 — Guts Round — Problem 10

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

[5] Let Pbe a polynomial such that P(x) =P(0) + P(1)x+P(2)x2andP(−1) = 1. Compute P(3).

解析

英文解析

  1. [ 5 ] Let P be a polynomial such that P ( x ) = P (0) + P (1) x + P (2) x and P ( − 1) = 1. Compute P (3).
    Answer: 5 Plugging in x = − 1 , 1 , 2 results in the trio of equations 1 = P ( − 1) = P (0) − P (1)+ P (2),
    P (1) = P (0) + P (1) + P (2) ⇒ P (1) + P (2) = 0, and P (2) = P (0) + 2 P (1) + 4 P (2). Solving theseas a system of equations in P (0) , P (1) , P (2) gives P (0) = − 1 , P (1) = − 1 , P (2) = 1. Consequently,
    P ( x ) = x − x − 1 ⇒ P (3) = 5.2