x-1/x=4,且x>0,則x^3+1/x^3=?

x - 1/x = 4
(x - 1/x)^2 = 4^2
x^2 - 2x(1/x) + (1/x)^2 = 16
x^2 + (1/x)^2 = 16 + 2x(1/x)
x^2 + (1/x)^2 = 18......(1)
x^2 + (1/x)^2 + 2x(1/x) = 18 + 2x(1/x)
x^2 + (1/x)^2 + 2x(1/x) = 20
(x + 1/x)^2 = 20
x + 1/x = +/- √20
x + 1/x = +/- 2√5..........(2)
而
x^3 + 1/x^3 = (x + 1/x)(x^2 - (x^3)(1/x^3) + 1/x^2)
= (x + 1/x)(x^2 + 1/x^2 - 1) , 代入(1)(2)結果 :
= (+/- 2√5)(18 - 1)
= +/- 34√5

2010-03-10 19:28:31 補充：
因 x > 0 ,

x^3 + 1/x^3 > 0

所以 - 34√5 捨去

答案 : x^3 + 1/x^3 = 34√5

2010-03-10 21:29:43 補充：
x^3 + 1/x^3 = (x + 1/x)(x^2 - (x^3)(1/x^3) + 1/x^2)

更正為

x^3 + 1/x^3 = (x + 1/x)(x^2 - (x)(1/x) + 1/x^2)

背錯式。

a^3 + b^3 = (a+b)(a^2 - ab + b^3)