A generic polynomial function has the formula/template
P(x)=anxn+an−1xn+⋯+a1x+a0.
A polynomial is called monic if an=1.
Suppose P has roots r1,r2,…,rn and derivative P′(x).
Then ri 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
P(x)P′(x)=x−r11+x−r21+⋯+x−rn1.
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∈S[x].
Specifically Z[x] and Q[x] and R[x] and 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]
is called irreducible over Z[x]
if there do not exist polynomials Q(x),R(x)∈Z[x]
different from ±1 such that P(x)=Q(x)R(x).
Otherwise P(x) is called reducible.
For a polynomial P(x)∈Z[x],
suppose that there exists a prime number p
such that an is not divisible by p,
ak is divisible by p for each k∈{0,1,…,n−1},
and a0 is not divisible by p2.
Then P(x) is irreducible over Z[x].