The inverse is a root of the reciprocal (reverse) polynomial. Let $F$ be a field and let $a \neq 0$ be a zero of the polynomial $a_0 + a_1x + . . . +a_nx^n$ in $F[x]$.
I want to show that $\frac{1}{a}$ is a zero of the polynomial $a_n + a_{n-1}x + . . . + a_0x^n$
How can I do this?

Hint $ $ The second reversed (aka reciprocal) polynomial is simply $\ \hat f(x)= x^n f(1/x),\,\ n = \deg f.\,$ Now verify that $\hat f(1/a) = 0\,$ since $\,f(a) = 0.$
