HMMT 十一月 2011 · 团队赛 · 第 3 题

HMMT November 2011 — Team Round — Problem 3

[ 6 ] Find the sum of the coefficients of the polynomial P ( x ) = x − 29 x + ax + bx + c , given that P (5) = 11, P (11) = 17, and P (17) = 23. 2

解析

[ 6 ] Find the sum of the coefficients of the polynomial P ( x ) = x − 29 x + ax + bx + c , given that P (5) = 11, P (11) = 17, and P (17) = 23. 4 3 2 Answer: -3193 Define Q ( x ) = P ( x ) − x − 6 = x − 29 x + ax + ( b − 1) x + ( c − 6) and notice that Q (5) = Q (11) = Q (17) = 0. Q ( x ) has degree 4 and by Vieta’s Formulas the sum of its roots is 29, so its last root is 29 − 17 − 11 − 5 = − 4, giving us Q ( x ) = ( x − 5)( x − 11)( x − 17)( x + 4). This means that P (1) = Q (1) + 7 = ( − 4)( − 10)( − 16)(5) + 7 = − 3200 + 7 = − 3193. 2