HMMT 十一月 2009 · 冲刺赛 · 第 14 题

HMMT November 2009 — Guts Round — Problem 14

[ 8 ] Let f ( x ) = x + ax + bx + cx + d be a polynomial whose roots are all negative integers. If a + b + c + d = 2009, find d . 2 2 2

解析

[ 8 ] Let f ( x ) = x + ax + bx + cx + d be a polynomial whose roots are all negative integers. If a + b + c + d = 2009, find d . Answer: 528 Call the roots − x , − x , − x , and − x . Then f ( x ) must factor as ( x + x )( x + x )( x + 1 2 3 4 1 2 x )( x + x ). If we evaluate f at 1, we get (1 + x )(1 + x )(1 + x )(1 + x ) = a + b + c + d + 1 = 3 4 1 2 3 4 2009 + 1 = 2010. 2010 = 2 · 3 · 5 · 67. d is the product of the four roots, so d = ( − 1) · ( − 2) · ( − 4) · ( − 66). 2 2 2