This is an exercise from Linear Algebra by Friedberg, Insel.
The question asks to determine if True or False: Any polynomial of degree $n$ with leading coefficient $(-1)^n$ is the characteristic polynomial of some linear operator.
I know that it is true after scouring the internet and trying to understand how to prove the result, but this is where I'm having trouble. From the different material I've read it says that the result can be proven by induction. How is the induction actually working? The following is where I run astray:
Take a polynomial of degree $n$ with leading coefficient $(-1)^n$. That gives us $$(-1)^{n}(a_0 + a_{1}t + \dots a_{n}t^{n})$$.
Right now this is just a general polynomial. The only notions that stick out to me as being indicative of a polynomial being the characteristic polynomial of a linear operator is by
calculating the characteristic polynomial explicitly for the linear operator by the $\det(T-\lambda I)$ process
The Cayley Hamilton Theorem (which I just learned about in the chapter I'm working on): $f(T) = T_{0}$ where $f(t)$ is the characteristic polynomial and $T_{0}$ is the zero linear operator.
Out of those two ideas the one that seems it could be most viable would be the Cayley Hamilton, but that would mean I need to assume that the polynomial is a characteristic polynomial Which is what I'm trying to prove.
There was another question posted on this years ago: Given any polynomial of degree $n \geq 1$ we can have a corresponding matrix with characteristic polynomial equal to the given polynomial
but the hints in that refer to the companion matrix, something I haven't encountered before.
How could I do the induction proof correctly? Or better how should I set things up because I feel that's where the problem is.