Show that polynomial with non-zero constant term of nilpotent matrix is invertible

Question

So I got a polynomial: $f(x) = a_nx^n+...+a_1x^1+a_0$ and $a_0 \neq 0$

And a matrix $A$ such that $A^k = 0$ for some $k$. I have to prove that $f(A)$ is invertible.

So, I know there is a fact that $E+A$ and $E-A$ are both invertible if $A$ is nilpotent and I'am trying to use that by computing $(E+A)^n$ and getting polynomial $g(A)$ but I am stuck here in attempts to connect $g(A)$ and $f(A)$

Answer 1

Hint: $f(A)=a_0(E+A\cdot(\dots))$, and $A\cdot(\dots)$ is nilpotent.