If 2 = (x + 1 x ) , then 3 + 1 2 (x^3 + 1 x^3 ) =

Step-by-step solution

Given x + 1 x = 2 Cubing both sides, we get: (x + 1 x )^3 = (2 )^3 x^3 + 1 x^3 + 3 (x + 1 x ) = 8 ^3 Substituting x + 1 x = 2 : x^3 + 1 x^3 + 3(2 ) = 8 ^3 x^3 + 1 x^3 = 8 ^3 - 6 x^3 + 1 x^3 = -2(3 - 4 ^3 ) Using the identity 3 = 3 - 4 ^3 : x^3 + 1 x^3 = -2 3 Dividing by 2 : 1 2 (x^3 + 1 x^3 ) = - 3 Therefore, the given expression becomes: 3 + 1 2 (x^3 + 1 x^3 ) = 3 - 3 = 0 Answer: 0