若x2+x-1=0,求x14+1/x14的值.(是x的平方,求x的14次方加上x14次方分之一)

Question

若x2+x-1=0,求x14+1/x14的值.(是x的平方,求x的14次方加上x14次方分之一)

Answer 1

感觉题目应该是x2+x+1=0为已知条件,这样比较好做.
x14+1/x14=(X7+1/X7)2-2(x7*(1/x7)) (X的7次方加上X7次方分之一的和的平方,减去二者乘积的两倍)因为x2+x+1=1/x(x+1+1/x)=0所以x+1+1/x=0.所以x+1/x=-1x2+1/x2=(x+1/x)2-2=-1x4+1/x4=...=-1x8+1/x8=...=-1 (到这步为止是必须做的.)(x8+1/x8)(x+1/x)=x9+x7+1/x7+1/x9=1x7+1/x7=1-(x9+1/x9)(x4+1/x4)(x2+1/x2)=x6+x2+1/x2+1/x6=x6+1/x6-1=1x6+1/x6=2(x6+1/x6)(x+1/x)=x7+x5+1/x5+1/x7=-2x7+1/x7=-2-(x5+1/x5)=1-(x9+1/x9)所以(x9+1/x9)-(x5+1/x5)=3