If α ,   β and γ are the roots of the equation x 3 - 3 x 2 + 4 x + 4 = 0 , then the value of α 2 + 1 1 1 1 β 2 + 1 1 1 1 γ 2 + 1 is equal to

Step-by-step solution

α + β + γ = 3 α β + β γ + γ α = 4 α β γ = - 4 α 2 + 1 1 1 1 β 2 + 1 1 1 1 γ 2 + 1 = α 2 + 1 β 2 + 1 γ 2 + 1 - 1 - 1 γ 2 + 1 - β 2 = α 2 + 1 β 2 + γ 2 + β 2 γ 2 - γ 2 - β 2 = α 2 β 2 + α 2 γ 2 + α 2 β 2 γ 2 + β 2 + γ 2 + β 2 γ 2 - γ 2 - β 2 = α 2 β 2 + β 2 γ 2 + γ 2 α 2 + α β γ 2 = α β + β γ + γ α 2 - 2 α β γ α + β + γ + α β γ 2 = 16 + 8 3 + 16 = 56