If x=a^2/3-a^-2/3 the find x^3+3x

Answer 1

x^3+3x = x(x^2+3)
= (a^2/3-a^-2/3)((a^2/3-a^-2/3)^2+3)
= (a^2/3-a^-2/3)(a^4/3 + 1 + a^-4/3)
Note we have the factorization of the difference of two cubes:
= a^2 - a^-2