HMMT 二月 2004 · 代数 · 第 10 题

英文解析

10. There exists a polynomial P of degree 5 with the following property: if z is a complex
    5 2
    number such that z +2004 z = 1, then P ( z ) = 0. Calculate the quotient P (1) /P ( − 1).
    Solution: − 2010012 / 2010013
    Let z , . . . , z be the roots of Q ( z ) = z +2004 z − 1. We can check these are distinct (by 5
    1 5
    using the fact that there’s one in a small neighborhood of each root of z + 2004 z , or by 5
    noting that Q ( z ) is relatively prime to its derivative). And certainly none of the roots
    5 5
    of Q is the negative of another, since z + 2004 z = 1 implies ( − z ) + 2004( − z ) = − 1,
    2 2
    so their squares are distinct as well. Then, z , . . . , z are the roots of P , so if we write
    1 5
    C for the leading coefficient of P , we have
    2 2
    P (1) C (1 − z ) · · · (1 − z )
    1 5
    2 2=
    P ( − 1) C ( − 1 − z ) · · · ( − 1 − z )
    1 5
    [(1 − z ) · · · (1 − z )] · [(1 + z ) · · · (1 + z )]
    1 5 1 5
    [( i − z ) · · · ( i − z )] · [( i + z ) · · · ( i + z )]=
    1 5 1 5
    [(1 − z ) · · · (1 − z )] · [( − 1 − z ) · · · ( − 1 − z )]
    1 5 1 5
    [( i − z ) · · · ( i − z )] · [( − i − z ) · · · ( − i − z )]=
    1 5 1 5
    5 5
    (1 + 2004 · 1 − 1)( − 1 + 2004 · ( − 1) − 1)
    5 5=
    ( i + 2004 · i − 1)( − i + 2004 · ( − i ) − 1)
    (2004)( − 2006)
    ( − 1 + 2005 i )( − 1 − 2005 i )=
    2005 − 12 = −
    2005 + 12 = − 4020024 / 4020026 = − 2010012 / 2010013 .
    Alternative Solution: In fact, we can construct the polynomial P explicitly (up tomultiplication by a constant). We write P ( z ) as a polynomial in z ; it must use only 2
    even powers of z and be divisible by z +2004 z − 1, so we are inspired to try a difference 53
    of squares,
    2 5 5 5 2 2 2 4 2
    P ( z ) = ( z + 2004 z − 1)( z + 2004 z + 1) = ( z + 2004 z ) − 1 = z ( z + 2004) − 1 ,
    giving
    2 2
    P ( z ) = z ( z + 2004) − 1 .
    2 2
    Now plugging in z = 1 and z = − 1 rapidly gives (2005 − 1) / ( − 2005 − 1) as before. 4