Suppose we are given a polynomial e.g.
$$x^4+Ax^3+Bx^2+Cx+D,\tag1$$
and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to make a matrix with some variables, find its characteristic polynomial, then set some values (found by trial and error) to the "extraneous" (WRT order of the polynomial) variables so that the matrix looks the simple enough, doesn't have extraneous degrees of freedom, but its characteristic polynomial still has the necessary number of degrees of freedom. One such matrix I've been able to come up with for polynomial $(1)$ looks like this:
$$ \begin{pmatrix} 0&0&1&-(B+D+1)\\ \frac{AD+C}{B+D+1}&0&0&1\\ D&0&0&0\\ 1&1&0&-A \end{pmatrix}. \tag2$$
But it looks complicated and ad hoc. I guess there should be better matrices, maybe even such that each of the polynomial coefficients would appear as element of the matrix (not in combination with others).
So my question is: is there a general systematic way to construct such matrices for polynomials of any given order?
An example of such matrix is the companion matrix of the polynomial, which in this case is
$$\begin{pmatrix} 0 & 0 & 0 & -D\\ 1 & 0 & 0 & -C\\ 0 & 1 & 0 & -B\\ 0 & 0 & 1 & -A \end{pmatrix}.$$