If the cubic equation z 3 + a z 2 + b z + c = 0   ∀ a , b , c ∈ R ,   c ≠ 0 has a purely imaginary root, then (where i 2 = - 1 )

Step-by-step solution

Let, z = i α ⇒ c - α 2 a + i b α - α 3 = 0 α ≠ 0 ⇒ α 2 = b ⇒ c = a b And z = ± i b