HMMT 二月 2010 · CALC 赛 · 第 3 题

英文解析

3. [ 4 ] Let p be a monic cubic polynomial such that p (0) = 1 and such that all the zeros of p ( x ) are alsozeros of p ( x ). Find p . Note: monic means that the leading coefficient is 1.
   Answer: ( x + 1) A root of a polynomial p will be a double root if and only if it is also a root of 3
   ′ ′
   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 )
   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 = 03
   and c = 1. From p (0) = 1 we get p ( x ) = ( x + 1) .3
   ∑
   | cos( k ) |nk =1