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HMMT 二月 2012 · 冲刺赛 · 第 10 题

HMMT February 2012 — Guts Round — Problem 10

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  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).
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 these as a system of equations in P (0) , P (1) , P (2) gives P (0) = − 1 , P (1) = − 1 , P (2) = 1. Consequently, 2 P ( x ) = x − x − 1 ⇒ P (3) = 5.