A generic polynomial function has the formula/template \(P(x) = a_nx^n + a_{n\!-\!1}x^{n\!-\!1} + \dotsb + a_1x + a_0\,.\) A polynomial is monic if \(a_n = 1.\)
For a polynomial \(P\) let \(P'\) denote its (formal) derivative. The polynomials \(P\) and \(P'\) will share a common root if and only if that root has multiplicity greater than one in \(P.\)
This should be verified, but \[ \frac{P'(x)}{P(x)} = \frac{1}{x-r_1} + \frac{1}{x-r_2} + \dotsb + \frac{1}{x-r_n}\,. \]
The complex zeros of the derivative \(P'\) of a polynomial \(P\) lie in the convex hull of the zeros of \(P.\)
If the polynomial \(P\) has coefficients from a specific set of numbers, say \(S,\) we write that \(P \in S[x].\) Specifically \(\mathbf{Z}[x]\) and \(\mathbf{Q}[x]\) and \(\mathbf{R}[x]\) and \(\mathbf{C}[x],\) the rings of polynomials over the integers, rationals, reals, and complex numbers respectively, come up pretty often. A polynomial \(P \in \mathbf{Z}[x]\) is called reducible over \(\mathbf{Z}[x]\) if there exist non-constant polynomials \(Q, R \in \mathbf{Z}[x]\) such that \({P(x) = Q(x)R(x).}\) We say that \(P\) is irreducible if it is not reducible.
For a polynomial \(P \in \mathbf{Z}[x],\) suppose that there exists a prime number \(p\) such that \(a_n\) is not divisible by \(p,\) \(a_k\) is divisible by \(p\) for each \(k \in \{0, 1,\dotsc, n\!-\!1\},\) and \(a_0\) is not divisible by \(p^2.\) Then \(P\) is irreducible over \(\mathbf{Z}[x].\)