polynomiaali

Polynomiaali is a term used in algebra to denote a polynomial function or the polynomial ring generated by a fixed indeterminate over a ring. In this article, Polynomiaali refers to the classical object of polynomial arithmetic: a finite linear combination of powers of a variable with coefficients from a ring.

Formally, a Polynomiaali P over a ring R is written as P(x) = a0 + a1 x + … + an x^n, where a0, a1, …, an are elements of R and an ≠ 0. The set of all Polynomiaali over R is denoted R[x], which carries the structure of a ring under the usual addition and multiplication of polynomials. For multiple variables, the concept extends to R[x1, ..., xk].

Operations and basic properties: Addition and multiplication are performed by standard polynomial rules, and evaluation at an element r in R substitutes x with r to yield an element of R. If P and Q are nonzero, the degree of their product satisfies deg(PQ) = deg(P) + deg(Q) in the common case where R is an integral domain. The leading coefficient of P is the coefficient of the highest-degree term. Over a field, the polynomial ring R[x] has favorable factorization properties: every nonconstant polynomial can be factored into irreducibles, and roots can be studied in extension fields.

Applications and related concepts: Polynomiaali form the foundation of solving algebraic equations, interpolation, coding theory, and algebraic geometry. They connect to the broader ideas of polynomial rings, irreducible polynomials, and polynomial equations. See also polynomial ring, irreducible polynomial, and roots of polynomials.