Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders.

Divisibility

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R, it is said that f divides g if there exists a polynomial q in R such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotient can be computed using the Horner scheme.

Related Topics:
Commutative algebra - Integral domain - Horner scheme

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If F is a field and f and g are polynomials in F with g ≠ 0, then there exist unique polynomials q and r in F with

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: f = q , g + r

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

and such that the degree of r is smaller than the degree of g. The polynomials q and r are uniquely determined by f and g. This is called "division with remainder" or "polynomial long division" and shows that the ring F is a Euclidean domain.

Related Topics:
Polynomial long division - Euclidean domain

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Analogously, polynomial "primes" (more correctly, irreducible polynomials) can be defined which cannot be factorized into the product of two polynomials of lesser degree. It is not easy to determine if a given polynomial is irreducible. One can start by simply checking if the polynomial has linear factors. Then, one can check divisibility by some other irreducible polynomials. Eisenstein's criterion can also be used in some cases to determine irreducibility.

Related Topics:
Irreducible - Degree - Eisenstein's criterion

~ ~ ~ ~ ~ ~ ~ ~ ~ ~