structural:reformatting, stylistic:tone
Terminology

Let

be a nonzero polynomial with p m ≠ 0 {\displaystyle p_{m}\neq 0}

The constant term of p is p 0. {\displaystyle p_{0}.} It is zero in the case of the zero polynomial.

The degree of p, written deg(p) is m, {\displaystyle m,} the largest k such that the coefficient of X is not zero.

The leading coefficient of p is p m. {\displaystyle p_{m}.}

In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and the degree has been variously left undefined, defined to be −1, or defined to be a −∞.

A constant polynomial is either the zero polynomial, or a polynomial of degree zero.

A nonzero polynomial is monic if its leading coefficient is 1. {\displaystyle 1.}

Given two polynomials p and q, if the degree of the zero polynomial is defined to be − ∞, {\displaystyle -\infty,} one has

and, over a field, or more generally an integral domain,

It follows immediately that, if K is an integral domain, then so is K.

It follows also that, if K is an integral domain, a polynomial is a unit (that is, it has a multiplicative inverse) if and only if it is constant and is a unit in K.

Two polynomials are associated if either one is the product of the other by a unit.

Over a field, every nonzero polynomial is associated to a unique monic polynomial.

Given two polynomials, p and q, one says that p divides q, p is a divisor of q, or q is a multiple of p, if there is a polynomial r such that q = pr.

A polynomial is irreducible if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.