Question

已知x+1/x=3,求下列各式的值:(1) x^2+1/x^2;(2) x^3+1/x^3;(3) x^2/x^4+x^2+1

Answer 1

(1)
x^2+1/x^2
=(x+1/x)^2-2
=3^2-2
=7
(2)
x^3+1/x^3
(由立方和公式)
=(x+1/x)(x^2-1+1/x^2)
(根据(1)的结果，x^2+1/x^2=7)
=3*(7-1)
=18
(3)
x^2/(x^4+x^2+1)
(分子分母同时除以x^2)
=1/(x^2+1+1/x^2)
(x^2+1/x^2=7)
=1/(7+1)
=1/8