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HMMT 十一月 2012 · GEN 赛 · 第 2 题

HMMT November 2012 — GEN Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

[3] Let Q(x) =x2+ 2x+ 3, and suppose that P(x) is a polynomial such that
P(Q(x)) =x6+ 6x5+ 18x4+ 32x3+ 35x2+ 22x+ 8.
Compute P(2).

解析

英文解析

  1. [ 3 ] Let Q ( x ) = x + 2 x + 3, and suppose that P ( x ) is a polynomial such that
    6 5 4 3 2
    P ( Q ( x )) = x + 6 x + 18 x + 32 x + 35 x + 22 x + 8 .
    Compute P (2).
    Answer: 2 Note that Q ( − 1) = 2. Therefore, P (2) = P ( Q ( − 1)) = 1 − 6+18 − 32+35 − 22+8 = 2.