X^3+1/x^3=110 find x+1/x

Answer 1

first, factor
(x³+1/x³)
=(x+1/x)(x²-x*(1/x)+1/x²)
=(x+1/x)(x²-1+1/x²)
Further factoring gives
=(x+1/x)((x+1/x)²-2(x*(1/x)-1)
=(x+1/x)((x+1/x)²-3)

Set k=(x+1/x), then
(x³+1/x³)=k(k^2-3)=110
Solve for k and reject complex roots
k=5
=>
(x+1/x)=5 [plus 2 more complex roots].