HMMT 二月 2010 · CALC 赛 · 第 3 题

HMMT February 2010 — CALC Round — Problem 3

[ 4 ] Let p be a monic cubic polynomial such that p (0) = 1 and such that all the zeros of p ( x ) are also zeros of p ( x ). Find p . Note: monic means that the leading coefficient is 1. ∑ n | cos( k ) | k =1

解析

[ 4 ] Let p be a monic cubic polynomial such that p (0) = 1 and such that all the zeros of p ( x ) are also zeros of p ( x ). Find p . Note: monic means that the leading coefficient is 1. 3 Answer: ( x + 1) A root of a polynomial p will be a double root if and only if it is also a root of ′ ′ p . Let a and b be the roots of p . Since a and b are also roots of p , they are double roots of p . But p ′ ′ 2 can have only three roots, so a = b and a becomes a double root of p . This makes p ( x ) = 3 c ( x − a ) 3 for some constant 3 c , and thus p ( x ) = c ( x − a ) + d . Because a is a root of p and p is monic, d = 0 3 and c = 1. From p (0) = 1 we get p ( x ) = ( x + 1) . ∑ n | cos( k ) | k =1