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HMMT 二月 2009 · CALC 赛 · 第 4 题

HMMT February 2009 — CALC Round — Problem 4

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 4 ] Let P be a fourth degree polynomial, with derivative P , such that P (1) = P (3) = P (5) = P (7) = 0. Find the real number x 6 = 1 , 3 , 5 such that P ( x ) = 0.
解析
  1. [ 4 ] Let P be a fourth degree polynomial, with derivative P , such that P (1) = P (3) = P (5) = P (7) = 0. Find the real number x 6 = 1 , 3 , 5 such that P ( x ) = 0. 89 Answer: 11 ′ P (7) Solution: Observe that 7 is not a root of P . If r , r , r , r are the roots of P , then = 1 2 3 4 P (7) ( ) − 1 ( ) ∑ ∑ − 1 1 1 1 1 1 = 0. Thus r = 7 − = 7 + + + = 7 + 12 / 11 = 89 / 11. 4 i i 6 =4 7 − r 7 − r 6 4 2 i i