HMMT 十一月 2010 · 冲刺赛 · 第 24 题

HMMT November 2010 — Guts Round — Problem 24

[ 12 ] Let P ( x ) be a polynomial of degree at most 3 such that P ( x ) = for x = 1 , 2 , 3 , 4. What is 2 1+ x + x P (5)?

解析

[ 12 ] Let P ( x ) be a polynomial of degree at most 3 such that P ( x ) = for x = 1 , 2 , 3 , 4. What is 2 1+ x + x P (5)? Guts Round − 3 Answer: The forward difference of a polynomial P is ∆ P ( x ) = P ( x + 1) − P ( x ), which is a new 91 polynomial with degree reduced by one. Therefore, if we apply this operation three times we’ll get a constant function, and we can work back up to get a value of P (5). Practically, we create the following table of differences: 1 1 1 1 3 7 13 21 − 4 − 6 − 8 21 91 273 34 10 273 273 − 24 273 Then extend it to be the following table: 1 1 1 1 − 9 3 7 13 21 273 − 4 − 6 − 8 − 22 21 91 273 273 34 10 − 14 273 273 273 − 24 − 24 273 273 − 9 − 3 So our answer is = 273 91