If α , β and γ are the roots of the equation x 3 + x + 2 = 0 , then the equation whose roots are α - β α - γ , β - γ β - α and γ - α γ - β is

Step-by-step solution

Putting y = α - β α - γ , we get, y = α 2 - α β + γ + β γ Now, α + β + γ = 0 and α β γ = - 2 ⇒ y = α 2 - α - α + - 2 α = 2 α 2 - 2 α = 2 α 3 - 1 α Also, α is a root of the given equation. ⇒ α 3 = - α - 2 ⇒ y = 2 - α - 2 - 1 α = 2 - α - 3 α = - 2 - 6 α ⇒ y + 2 = - 6 α ⇒ α = - 6 y + 2 α is a root of the given equation ⇒ - 6 y + 2 3 + - 6 y + 2 + 2 = 0 - 216 - 6 y + 2 2 + 2 y + 2 3 = 0 ⇒ y + 2 3 - 3 y + 2 2 - 108 = 0 ⇒ y 3 + 6 y 2 + 12 y + 8 - 3 y 2 - 12 - 12 y - 108 = 0 ⇒ y 3 + 3 y 2 - 112 = 0 Hence, the required equ