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HMMT 二月 2007 · TEAM1 赛 · 第 13 题

HMMT February 2007 — TEAM1 Round — Problem 13

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 30 ] Find all nonconstant polynomials P ( x ), with real coefficients and having only real zeros, such that 2 3 P ( x + 1) P ( x − x + 1) = P ( x + 1) for all real numbers x .
解析
  1. [ 30 ] Find all nonconstant polynomials P ( x ), with real coefficients and having only real zeros, such that 2 3 P ( x + 1) P ( x − x + 1) = P ( x + 1) for all real numbers x . 5 k + Answer: { P ( x ) = x | k ∈ Z } . 3 Solution. Note that if P ( α ) = 0, then by setting x = α − 1 in the given equation, we find 0 = P ( x + 3 2