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

Step-by-step solution

Given = 1 2 (x+ 1 x ) , then 2 = x+ 1 x . The expression x^3+ 1 x^3 can be rewritten using the algebraic identity for cubes: x^3+ 1 x^3 = (x+ 1 x )^3 - 3 (x+ 1 x ) . Substituting x+ 1 x = 2 yields x^3+ 1 x^3 = 8 ^3 - 6 . The target expression becomes: E = 3 + 1 2 (x^3+ 1 x^3 ) = 3 +4 ^3 - 3 . Using the triple angle identity 3 = 3 - 4 ^3 , this simplifies to: E = (3 - 4 ^3 ) + 4 ^3 - 3 = 0 . The final result is 0 .