Question

已知x+1/x=1,则x^11+x^78+1/x^23+1/x^78=

Answer 1

(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