Irreducibility of the polynomial $a_{n}+a_{n-1}x+\cdots+a_{0}x^{n}$ when irreducibility of $a_{0}+a_{1}x+\cdots+a_{n}x^{n}$ is given

Question

Given that $\mathbb F$ is a field and $\mathbb F[x]$ is the polynomial ring over $\mathbb F$. $\ \ $If the polynomial $a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}$ is irreducible over $\mathbb F[x]$ then so is the polynomial $a_{n}+a_{n-1}x+a_{n-2}x^{2}+\cdots+a_{0}x^{n}$.

Please give me some hints as to how to begin to think the solution.

Thanks for any help.

Answer 1

Suppose that

$$a_n + a_{n-1} x + \ldots + a_0 x^n = \left( b_i + b_{i-1} x + \ldots + b_0 x^i \right) \left( c_j + c_{j-1} x + \ldots + c_0 x^j \right).$$

Then what is $\left( b_0 + b_1 x + \ldots + b_i x^i \right) \left( c_0 + c_1 x + \ldots + c_j x^j \right)$ ?