初中代数 急

来源：百度知道 编辑：UC知道 时间：2024/06/27 07:57:57
已知x^2+x+1=0
求有理式x^14+1/x^14的值

好恶心的题

x^2+x+1=0
x=0不是它的根
=>
x + 1 + 1/x = 0
=>
x + 1/x = -1
xx + 1/xx = (x + 1/x)^2 - 2 = -1
x^4 + 1/x^4 = (xx+1/xx)^2 - 2 = -1
同理：
x^8 + 1/x^8 = -1
x^16 + 1/x^16 = -1
x^16 + 1/x^16 = (xx+1/xx)*(x^14 + 1/x^14) - (x^12+1/x^12)

已知：
x^3 + 1/x^3 = (x+1/x)^3 - 3(x+1/x)=2
=>
x^6 + 1/x^6 = (x^3 + 1/x^3)^2 - 2 = 2
=>
x^12 + 1/x^12 = 2
代入得
-1 = -1*(x^14 + 1/x^14) - 2
=>
(x^14 + 1/x^14)=-1