已知x²-5x-1997=0,求代数式[（x-2）³-（x-1）²+1]/(x-2)

已知x²-5x-1997=0,求代数式[（x-2）³-（x-1）²+1]/(x-2)
已知x²-5x-1997=0,求代数式[（x-2）³-（x-1）²+1]/(x-2)

已知x²-5x-1997=0,求代数式[（x-2）³-（x-1）²+1]/(x-2)
∵x²-5x-1997=0
∴x²=5x+1997
∴[(x-2)³-(x－1)²＋1]/(x-2)
=(x-2)²-[(x-1)²-1]/(x-2)
又(x-2)²=x²-4x+4=5x+1997-4x+4=x+2001
(x-1)²-1=x²-2x=5x+1997-2x=3x+1997
∴原式=x+2001-(3x+1997)/(x-2)
=2001+(x²-2x-3x-1997)/(x-2)
=2001+(x²-5x-1997)/(x-2)
=2001+0
=2001