Products of a polynomial and its conjugate

Question

I was wondering if, given a polynomial $$f(x)= a_n x^n + \dots + a_0 \in \mathbb{C}[x]$$ and its complex conjugate $$\overline{f(x)}= \overline{a_n}x^n + \dots + \overline{a_0} \in \mathbb{C}[x],$$ the products $$g(x) = f(x) \overline{f(x)}$$ has real coefficients. Is there a simple way to prove this?

Thank you!

Answer 1

Yes, it is true. The $k$^th coefficient is$$a_0\overline{a_k}+a_1\overline{a_{k-1}}+\cdots+a_k\overline{a_0},$$which is real beacause it is equal to its own conjugate.

Answer 2

The hint:

Prove that $$\overline{g}=g$$

Answer 3

Hint

$$\lambda \in\mathbb R\iff \bar \lambda =\lambda .$$

Answer 4

Yes. It's clear that $g(x)\in\Bbb R$ for all $x\in\Bbb R$; hence the coefficients of $g$ are real, since they're given by derivatives of $g$ at the origin.