Question

已知x^2-3x-1=0,求(x^10+x^8+x^2+1)/(x^10+x^6+x^4+1)的值

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Answer 1

设t=x²+1/x², 则t²-2=x^4+1/x^4
由x²-3x-1=0 得 x-1/x=3, (x-1/x)²=9
x²+1/x²=11 即 t=11
(x^10+x^8+x²+1)/(x^10+x^6+x^4+1)
=(x²+1)(x^8+1)/((x^6+1)(x^4+1))
=(x²+1)(x^8+1)/((x²+1)(x^4-x²+1)(x^4+1))
=(x^8+1)/((x^4-x²+1)(x^4+1))
=(x^4+1/x^4)/((x²+1/x²-1)(x²+1/x²))
=(t²-2)/((t-1)·t)
=(11²-2)/((11-1)·11)
=119/110