HMMT 十一月 2008 · 冲刺赛 · 第 29 题

HMMT November 2008 — Guts Round — Problem 29

[ 14 ] Let p ( x ) be the polynomial of degree 4 with roots 1 , 2 , 3 , 4 and leading coefficient 1. Let q ( x ) be p ( x ) 1 1 1 the polynomial of degree 4 with roots 1 , , , and and leading coefficient 1. Find lim . x → 1 2 3 4 q ( x )

解析

[ 14 ] Let p ( x ) be the polynomial of degree 4 with roots 1 , 2 , 3 , 4 and leading coefficient 1. Let q ( x ) be p ( x ) 1 1 1 the polynomial of degree 4 with roots 1 , , , and and leading coefficient 1. Find lim . x → 1 2 3 4 q ( x ) ( 4 Answer: − 24 Consider the polynomial f ( x ) = x q ( )( x ) – it has the same roots, 1, 2, 3, and 4, 1 as p ( x ). But this polynomial also has the same coefficients as q ( x ), just in reverse order. Its leading 1 1 1 1 1 coefficient is q (0) = 1 · · · = . So f ( x ) is p ( x ) scaled by , which means that p ( x ) /f ( x ) goes 2 3 4 24 24 to 24 as x goes to 1, and f ( x ) /q ( x ) goes to − 1.