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HMMT 二月 2001 · 代数 · 第 7 题

HMMT February 2001 — Algebra — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. A polynomial P has four roots, , , 2 , 4. The product of the roots is 1, and P (1) = 1. 4 2 Find P (0).
解析
  1. A polynomial P has four roots, , , 2 , 4. The product of the roots is 1, and P (1) = 1. 4 2 Find P (0). Solution: A polynomial Q with n roots, x , . . . , x , and Q ( x ) = 1 is given by Q ( x ) = 1 n 0 ( x − x )( x − x ) ··· ( x − x ) 1 2 n 1 8 , so P (0) = = . 3 1 ( x − x )( x − x ) ··· ( x − x ) 9 · · ( − 1) · ( − 3) 0 1 0 2 0 4 4 2