Let α ,   β and γ be the roots of the equation x 3 + x + 1 = 0 , then α β α + β + β γ β + γ + γ α γ + α α 2 + β 2 + γ 2 is equal to

Step-by-step solution

α + β + γ = 0 , ∑ α β = 1 , α β γ = - 1 ∑ α β α + β = ∑ α β - γ = - 3 α β γ = 3 α 2 + β 2 + γ 2 = α + β + γ 2 - 2 α β + β γ + γ α = - 2 Hence, ∑ α β α + β α 2 + β 2 + γ 2 = - 3 2