若x-1\x=根5,则(x^10+x^6+x^4+1)\(x^10+x^8+x^2+1)的值为

Question

若x-1\x=根5,则(x^10+x^6+x^4+1)\(x^10+x^8+x^2+1)的值为

Answer 1

x-1/x=√5
（x-1/x）ˇ2=5
xˇ2+1/xˇ2=5-2=3
xˇ4+1=3xˇ2,
xˇ4+1/xˇ4=7
xˇ8+1=7xˇ4
(x^10+x^6+x^4+1)\(x^10+x^8+x^2+1)
=[x^6(x^4+1)+(x^4+1)]/[x^8(x^2+1)+(x^2+1)]
=(x^6+1)(x^4+1)/(x^8+1)(x^2+1)
=3xˇ2（xˇ6+1)/7xˇ4(xˇ2+1）
=3（xˇ2+1）(xˇ4-xˇ2+1）/7xˇ2（xˇ2+1）
=3xˇ2/7xˇ2
=3/7