已知x+1/x=1,则x^11+x^78+1/x^23+1/x^78=

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已知x+1/x=1,则x^11+x^78+1/x^23+1/x^78=
已知x+1/x=1,则x^11+x^78+1/x^23+1/x^78=

已知x+1/x=1,则x^11+x^78+1/x^23+1/x^78=
(x+1/x)^2=1 拆开得 x^2+2+1/x^2=1 即 x^2+1/x^2=-1
x^2+1=x
x^2=x-1
x^3=x^2-x=(x-1)-x=-1
x^3+1=0 且x≠-1
x^11+x^78+1/x^23+1/x^78
=-x^2+1-1/x^2+1
=-(x^2+1/x^2)+2
=3