Q13/080) Let α +i β : α and β are real , be a root of the equation x^3 + qx +r = 0 , q and r are real. Find a real cubic equation, independent of α and β whose one root is 2 α.

As q and r are real and  α +i β is a root so  α - i β is a root.
Now sum of roots = (coefficient of x^2  =) 0)

Ler 3^rd root be γ, so γ + (α +i β) + (α- i β) = γ + 2 α = 0

Or γ = -  2 α

So -  2 α is root of f(x) = x^3 + qx +r  =0

So   2 α is root of f(-x) = - x^3 – q x + r = 0 or x^3 + qx – r = 0

equation  x^3 + qx  - r = 0