Let \(k\) be a field, \(A \in \Mat_{n\times n}(k)\), and \(\chi_A(t) \coloneqq \det(tI - A)\). Then \[ \begin{aligned} \chi_A(t) &= \sum_{k=0}^n (-1)^k \trace\qty{\Extpower^k A}\, t^{n-k} \\ &= t^n - \trace\qty{A} t^{n-1} + \trace\qty{\Extpower^2 A}\, t^{n-2} - \cdots + (-1)^{n-1} \trace\qty{\Extpower^{n-1} A}\, t + (-1)^n \det(A), \end{aligned} \] where \(\trace\qty{\Extpower^0 A} = 1\), \(\Extpower^1 A = A\), and \(\trace\qty{\Extpower^n A} = \det(A)\). Moreover, for \(0 \leq \ell \leq n\), \[ \trace\qty{\Extpower^\ell A} = \sum_{\substack{S \subseteq \theset{1, \ldots, n} \\ \abs{S} = \ell}} \det\qty{A_{S,S}}, \] the sum of the \(\binom{n}{\ell}\) principal \(\ell\times\ell\) minors of \(A\), where \(A_{S,S}\) is the submatrix of \(A\) with rows and columns indexed by \(S\), obtained by deleting the rows and columns indexed by the complement of \(S\).
Let \(M = (4i + j)_{0 \leq i, j \leq 3} \in \Mat_{4\times 4}(\QQ)\). The principal submatrices of \(M\) of sizes \(\ell=1,\ldots,4\):
The \(1\times 1\) minors sum to \(\trace M = 0 + 5 + 10 + 15 = 30\), and the \(2\times 2\) minors are \(-4, -16, -36, -4, -16, -4\), with sum \(-80\). Every row of \(M\) is a linear combination of \((1,1,1,1)\) and \((0,1,2,3)\), so \(\rank M = 2\) and every \(3\times 3\) and \(4\times 4\) minor vanishes. Therefore \(\chi_M(t) = t^4 - 30t^3 - 80t^2\).
Let \(f(t) = a_n t^n + \cdots + a_1 t + a_0 \in \ZZ[t]\) with \(a_n \neq 0\). If \(p/q \in \QQ\) is a root of \(f\) with \(p, q \in \ZZ\) coprime, then \(p \divides a_0\) and \(q \divides a_n\). In particular, if \(f\) is monic, every rational root of \(f\) is an integer dividing \(a_0\).
If \(r\) is a root of \(f\), polynomial long division gives \(f(t) = (t - r)\,g(t)\) with \(\deg g = \deg f - 1\), and the remaining roots of \(f\) are the roots of \(g\).