HMMT 二月 2006 · CALC 赛 · 第 1 题
HMMT February 2006 — CALC Round — Problem 1
题目详情
英文原题
- 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 xe − 1 − x
解析
英文解析
- 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 )?
Answer:1
n 18
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, son n − 1 n − 2
cx = cnx · cn ( n − 1) x , which implies that n = ( n − 1) + ( n − 2), or n = 3, andc = cn · cn ( n − 1) = 18 c , or c = .21
x cos x 18
e − 1 − x