HMMT 二月 2007 · 代数 · 第 10 题

HMMT February 2007 — Algebra — Problem 10

[ 8 ] The polynomial f ( x ) = x +17 x +1 has distinct zeroes r , . . . , r . A polynomial P of degree 1 2007 ( ) 1 2007 has the property that P r + = 0 for j = 1 , . . . , 2007. Determine the value of P (1) /P ( − 1). j r j 1

解析

[ 8 ] The polynomial f ( x ) = x +17 x +1 has distinct zeroes r , . . . , r . A polynomial P of degree 1 2007 ( ) 1 2007 has the property that P r + = 0 for j = 1 , . . . , 2007. Determine the value of P (1) /P ( − 1). j r j 289 Answer: . For some constant k , we have 259 ( ( )) 2007 ∏ 1 P ( z ) = k z − r + . j r j j =1 3 2 Now writing ω = 1 with ω 6 = 1, we have ω + ω = − 1. Then “ “ ”” Q 2007 1 2 k 1 − r + 2 ∏ ∏ j j =1 r − r +1 r 2007 j 2007 ( − ω − r )( − ω − r ) j j j j “ “ ”” P (1) /P ( − 1) = = = Q 2 2 j =1 j =1 2007 1 ( ω − r )( ω − r ) r + r +1 j j j k − 1 − r + j j j =1 r j 2007 2006 2 2007 2 2006 2 2 − ω +17 ω +1 − ( ω ) +17( ω ) +1 ( )( ) f ( − ω ) f ( − ω ) (17 ω )(17 ω ) = = = 2 2007 2006 2 2007 2 2006 2 f ( ω ) f ( ω ) ( ω +17 ω +1)(( ω ) +17( ω ) +1) (2+17 ω )(2+17 ω ) 289 289 = = . 2 4+34( ω + ω )+289 259 3