HMMT 十一月 2008 · 冲刺赛 · 第 19 题

HMMT November 2008 — Guts Round — Problem 19

[ 11 ] Let P be a polynomial with P (1) = P (2) = · · · = P (2007) = 0 and P (0) = 2009!. P ( x ) has leading coefficient 1 and degree 2008. Find the largest root of P ( x ).

解析

[ 11 ] Let P be a polynomial with P (1) = P (2) = · · · = P (2007) = 0 and P (0) = 2009!. P ( x ) has leading coefficient 1 and degree 2008. Find the largest root of P ( x ). Answer: 4034072 P (0) is the constant term of P ( x ), which is the product of all the roots of the polynomial, because its degree is even. So the product of all 2008 roots is 2009! and the product of 2009! the first 2007 is 2007!, which means the last root is = 2009 · 2008 = 4034072. 2007!