HMMT 二月 2006 · CALC 赛 · 第 1 题

HMMT February 2006 — CALC Round — Problem 1

A nonzero polynomial f ( x ) with real coefficients has the property that f ( x ) = f ( x ) f ( x ). What is the leading coefficient of f ( x )? x cos x e − 1 − x

解析

A nonzero polynomial f ( x ) with real coefficients has the property that f ( x ) = f ( x ) f ( x ). What is the leading coefficient of f ( x )? 1 Answer: 18 n Solution: Suppose that the leading term of f ( x ) is cx , where c 6 = 0. Then the ′ ′′ n − 1 n − 2 leading terms of f ( x ) and of f ( x ) are cnx and cn ( n − 1) x , respectively, so n n − 1 n − 2 cx = cnx · cn ( n − 1) x , which implies that n = ( n − 1) + ( n − 2), or n = 3, and 1 2 c = cn · cn ( n − 1) = 18 c , or c = . 18 x cos x e − 1 − x