HMMT 二月 2008 · 冲刺赛 · 第 13 题

HMMT February 2008 — Guts Round — Problem 13

[ 8 ] Let P ( x ) be a polynomial with degree 2008 and leading coefficient 1 such that P (0) = 2007 , P (1) = 2006 , P (2) = 2005 , . . . , P (2007) = 0 . Determine the value of P (2008). You may use factorials in your answer. ∑ ∞ n

解析

[ 8 ] Let P ( x ) be a polynomial with degree 2008 and leading coefficient 1 such that P (0) = 2007 , P (1) = 2006 , P (2) = 2005 , . . . , P (2007) = 0 . Determine the value of P (2008). You may use factorials in your answer. Answer: 2008! − 1 Consider the polynomial Q ( x ) = P ( x ) + x − 2007. The given conditions tell us that Q ( x ) = 0 for x = 0 , 1 , 2 , . . . , 2007, so these are the roots of Q ( x ). On the other hand, we know that Q ( x ) is also a polynomial with degree 2008 and leading coefficient 1. It follows that Q ( x ) = x ( x − 1)( x − 2)( x − 3) · · · ( x − 2007). Thus P ( x ) = x ( x − 1)( x − 2)( x − 3) · · · ( x − 2007) − x + 2007 . Setting x = 2008 gives the answer. ∑ ∞ n