HMMT 二月 2004 · 代数 · 第 10 题
HMMT February 2004 — Algebra — Problem 10
题目详情
- 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). 1
解析
- 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 5 Let z , . . . , z be the roots of Q ( z ) = z +2004 z − 1. We can check these are distinct (by 1 5 5 using the fact that there’s one in a small neighborhood of each root of z + 2004 z , or by 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 ) 2 2005 − 1 = − 2 2005 + 1 = − 4020024 / 4020026 = − 2010012 / 2010013 . Alternative Solution: In fact, we can construct the polynomial P explicitly (up to 2 multiplication by a constant). We write P ( z ) as a polynomial in z ; it must use only 3 5 even powers of z and be divisible by z +2004 z − 1, so we are inspired to try a difference 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