Complex number quadratic equation excercise

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.

Re: Complex number quadratic equation excercise

The problem stems from the fact that you have a linear polynomial with real coefficients and you want a complex (non-real) solution. If $\displaystyle k \neq 0$ then $\displaystyle \lambda$ must be a complex number in order for $\displaystyle 1+i\sqrt{3}$ to be a root of $\displaystyle kz+\lambda$.

That said, what you've written is good. You've shown that the only way for this complex number to be a root of this real polynomial is for both the coefficients to be 0. Now, I suspect you are actually supposed to have a quadratic polynomial.

Re: Complex number quadratic equation excercise

Indeed this was the second part of the exercise, in the first part I was given polynomial f(z) = http://latex.codecogs.com/gif.latex?z^{2} - 2z + 3 where z belongs to C and was asked to solve f(z) = 0. z1 was 1 + i√2 and z2 was 1 - i√2 and since in the second part I was told 1 + i√3 is the root I thought f(z) now was a different polynomial ( κz + λ ).

Also I noticed you use codecogs equation editor, is it integrated somewhere in this forum or do you use the standalone and then copy the gifs ?

Re: Complex number quadratic equation excercise

It's integrated. Write TEX /TEX with square brackets "[", "]" around each to make it act like $$ in LaTeX.