Given the fact that 1 + i√3 is the root of f(z) = κz + λ where κ,λ belong to R I am asked to find κ and λ. I already know the correct way to solve this but I would like to know why my approach is incorrect.

Heres what I thought of,

(1 + i√3)κ + λ = 0

κ + κi√3 + λ = 0

(κ + λ) + (κ√3)i = 0

(κ + λ) = 0 and (κ√3) = 0

κ = 0 and λ = 0

PS: Forgive me if I explained and/or presented anything the wrong way, I was taught this in greek and my english math vocabulary is quite poor. Thanks in advance.