x+1/x = -5 what is x^3+1/(x^3)
If x + 1/x = -5 then squaring both sides, we have that
(x + 1/x)^2 = 25 → x^2 + 2 + 1/x^2 = 25 → x^2 + 1/x^2 = 23
Note that x^3 + 1/ (x^3) can be factored as a sum of cubes thusly :
(x + 1/x) ( x^2 - x(1/x) + 1/x^2 ) =
(x + 1/x) ( x^2 - 1 + 1/x^2 ) =
(x + 1/x) ( x^2 + 1/x^2 - 1 ) substituting
( -5) ( 23 - 1 ) =
(-5) (22) =
-110