Question

已知x^2-3x+1=0，求x^2/(x^4+3x^2+1）及x^3+1/(x^3)的值。急！！！

Answer 1

解：
x²-3x+1=0
x- 3 +1/x=0
x+ 1/x=3

x²/(x⁴+3x²+1)
=1/(x²+3 +1/x²)
=1/(x²+2+ 1/x² +1)
=1/[(x+ 1/x)²+1]
=1/(3²+1)
=1/10

x³+ 1/x³
=(x+ 1/x)(x² -x ·1/x +1/x²)
=(x+ 1/x)(x²+2x·1/x+ 1/x² -3x·1/x)
=(x+ 1/x)[(x+ 1/x)²-3]
=3/(3²-3)
=1/2

追问

X^3+1/x^3的最后两步好像不对吧~

追答

嗯，是不对。重新写一下：
x³+ 1/x³
=(x+ 1/x)(x² -x ·1/x +1/x²)
=(x+ 1/x)(x²+2x·1/x+ 1/x² -3x·1/x)
=(x+ 1/x)[(x+ 1/x)²-3]
=3·(3²-3)
=18

Answer 2