Proof of Complex Conjugates and Real Coefficients | Complex Numbers Homework

[astrololo]

Homework Statement

I have two complex numbers that are non real, k and z. K and z are going to be complex conjugates if and only if the product (x-k)(x-z) is a polynomial with real coefficients.

Here is my answer :

k=a+bi

z=c+di

(x-k)(x-z) = x^2 -(k+z)x+kz

Homework Equations

The Attempt at a Solution

I was able to prove that a=c and d=-b (I have proven they're conjugatCes)

But because this is a if and only if, I must prove that if they're conjugates, then the coefficients are real. How do I do that ?

 

[BvU]

So k = a + bi and z = a - bi. Now write out (x-k)(x-z) and see if there's anything complex left over or not !

 

[astrololo]

So k = a + bi and z = a - bi. Now write out (x-k)(x-z) and see if there's anything complex left over or not !

To be honest, I did think about doing that but I was lazy and didn't try it and went just to ask this question... Thank you !

 

[BvU]

Being lazy is often a good quality for finding an economic way out of a problem