Polynomials

A generic polynomial function has the formula/template P(x)=anxn+an1xn++a1x+a0.P(x) = a_nx^n + a_{n-1}x^n + \dotsb + a_1x + a_0. A polynomial is called monic if an=1.a_n = 1.

Suppose PP has roots r1,r2,,rnr_1, r_2, \dotsc, r_n and derivative P(x).P'(x). Then rir_i will also be a root of P(x)P'(x) if and only if it is a root of multiplicity greater than one of P(x)P(x).

This should be verified, but P(x)P(x)=1xr1+1xr2++1xrn. \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)P'(z) of a polynomial P(z)P(z) lie in the convex hull of the zeros of P(z).P(z).

If the polynomial PP has coefficients from a specific set of numbers, say S,S, we write that PS[x].P \in S[x]. Specifically Z[x]\mathbf{Z}[x] and Q[x]\mathbf{Q}[x] and R[x]\mathbf{R}[x] and C[x],\mathbf{C}[x], the rings of polynomials over the integers, rationals, reals, and complex numbers respectively, come up pretty often. A polynomial P(x)Z[x]P(x) \in \mathbf{Z}[x] is called irreducible over Z[x]\mathbf{Z}[x] if there do not exist polynomials Q(x),R(x)Z[x]Q(x), R(x) \in \mathbf{Z}[x] different from ±1\pm 1 such that P(x)=Q(x)R(x).{P(x) = Q(x)R(x).} Otherwise P(x)P(x) is called reducible.

For a polynomial P(x)Z[x],P(x) \in \mathbf{Z}[x], suppose that there exists a prime number pp such that ana_n is not divisible by p,p, aka_k is divisible by pp for each k{0,1,,n1},k \in \{0, 1,\dotsc, n-1\}, and a0a_0 is not divisible by p2.p^2. Then P(x)P(x) is irreducible over Z[x].\mathbf{Z}[x].