The inverse is a root of the reciprocal (reverse) polynomial.
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 $\,f(a) = 0.$
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 $\,f(a) = 0.$