The sum of the cubes of all the roots of the equation x 4 - 3 x 3 - 2 x 2 + 3 x + 1 = 0 is _____.

Step-by-step solution

x 4 - 3 x 3 - 2 x 2 + 3 x + 1 = 0 Dividing by x 2 as x ≠ 0 x 2 - 3 x - 2 + 3 x + 1 x 2 = 0 x 2 + 1 x 2 - 2 + 2 - 3 x - 1 x - 2 = 0 x - 1 x 2 - 3 x - 1 x = 0 x - 1 x = 0 , x - 1 x = 3 x 2 - 1 = 0 , x 2 - 3 x - 1 = 0 x = ± 1 For x 2 - 3 x - 1 = 0 let the roots are γ   &   δ α = 1 , β = - 1 ,   γ + δ = 3 , γ δ = - 1 α 3 + β 3 + γ 3 + δ 3 = 1 - 1 + γ + δ γ + δ 2 - 3 γ δ = 3 9 - 3 - 1 = 36