HMMT 二月 2007 · 代数 · 第 10 题

英文解析

10. [ 8 ] The polynomial f ( x ) = x +17 x +1 has distinct zeroes r , . . . , r . A polynomial P of degree
    1 2007
    ( )
    2007 has the property that P r + = 0 for j = 1 , . . . , 2007. Determine the value of P (1) /P ( − 1).1
    289 jrj
    Answer: . For some constant k , we have
    ( ( ))259
    2007
    ∏
    P ( z ) = k z − r + .1
    j =1 jrj
    3 2
    Now writing ω = 1 with ω 6 = 1, we have ω + ω = − 1. Then
    “ “ ””
    2007Qk 1 − r + 221
    ∏ ∏
    j =1 r − r +1 jr
    2007 j 2007 ( − ω − r )( − ω − r )
    j j j j
    “ “ ””
    P (1) /P ( − 1) = = =
    2 2Qj =1 j =1
    2007 1 ( ω − r )( ω − r )
    r + r +1
    j jjk − 1 − r + jjj =1 rj
    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 = = .
    4+34( ω + ω )+289 2592 3