已知x²-16x-1=0,求x³-1\x³的值

已知x²-16x-1=0,求x³-1\x³的值
已知x²-16x-1=0,求x³-1\x³的值

已知x²-16x-1=0,求x³-1\x³的值
x²-16x-1=0
x-16-1/x=0
∴x-1/x=16
x³-1/x³=(x-1/x)(x²+1+1/x²)
=(x-1/x)[(x-1/x)²+3]
=16×(16²+3)
=4144

x²-1=16x
两边除以x
x-1/x=16
两边平方
x²-2+1/x²=256
x²+1/x²=258
所以x³-1/x³
=(x-1/x)(x²+1+1/x²)
=16*(258+1)
=4144

由x^2-16x-1=0得
x-1/x =16
(x^3-1/x^3)
=(x-1/x)(x^2+1+1/x^2)
=(x-1/x)((x-1/x)^2+3)
=16*(16^2+3)
=16^3 +48

x²-16x-1=0
x-1/x=16 x^2+1/x^2=16^2+2=258
x³-1\x³
=(x-1/x)(x^2+1+x*1/x+1/x^2)
=16*(258+1)
=16*250+16*9
=4000+144
=4144