A generic polynomial function has the formula/template \(P(x) = a_nx^n + a_{n-1}x^n + \dotsb + a_1x + a_0.\) A polynomial is called monic if \(a_n = 1.\)
Suppose \(P\) has roots \(r_1, r_2, \dotsc, r_n\) and derivative \(P'(x).\) Then \(r_i\) will also be a root of \(P'(x)\) if and only if it is a root of multiplicity greater than one of \(P(x)\).
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 \(P'(z)\) of a polynomial \(P(z)\) lie in the convex hull of the zeros of \(P(z).\)
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 \(P(x) \in \mathbf{Z}[x]\) is called irreducible over \(\mathbf{Z}[x]\) if there do not exist polynomials \(Q(x), R(x) \in \mathbf{Z}[x]\) different from \(\pm 1\) such that \({P(x) = Q(x)R(x).}\) Otherwise \(P(x)\) is called reducible.
For a polynomial \(P(x) \in \mathbf{Z}[x],\) 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 \(P(x)\) is irreducible over \(\mathbf{Z}[x].\)