If f ( x ) and g ( x ) are two polynomials such that the polynomial P ( x ) = f x 3 + x g x 3 is divisible by x 2 + x + 1 , then P ( 1 ) is equal to ___ .

Step-by-step solution

P ( x ) = f x 3 + x g x 3 P ( 1 ) = f ( 1 ) + g ( 1 )       . . . 1 Now P ( x ) is divisible by x 2 + x + 1 ⇒ P ( x ) = Q ( x ) x 2 + x + 1 P ( ω ) = 0 = P ω 2 where ω , ω 2 are non-real cube roots of units P ( x ) = f x 3 + x g x 3 P ( ω ) = f ω 3 + ω g ω 3 = 0 f ( 1 ) + ω g ( 1 ) = 0       . . . 2 P ω 2 = f ω 6 + ω 2 g ω 6 = 0 f ( 1 ) + ω 2 g ( 1 ) = 0       . . . 3 Now, ( 2 ) + ( 3 ) ⇒ 2