Polinomin

Polinomin is a term used in some mathematical texts to denote a polynomial—a finite linear combination of powers of one or more variables with coefficients in a ring or field. In one variable, a polinomin P over a ring R is written as P(x) = a_0 + a_1 x + ... + a_n x^n, with a_n ≠ 0. The degree of P is n and the leading coefficient is a_n. In several variables, P(x_1,...,x_k) = ∑ a_{i_1,...,i_k} x_1^{i_1} ... x_k^{i_k}, often restricted to total degree n.

Polinomins are studied within polynomial rings. The set R[x] of polinomins with coefficients in R forms a ring under addition and multiplication. If R is a field, F[x] is a principal ideal domain and supports division with remainder, factorization into irreducibles, and root-finding in extensions. Evaluation at a point, P(c), is a basic operation, and the factor theorem connects roots to linear factors. Over algebraically closed fields, a polinomin of degree n has exactly n roots counted with multiplicity in the algebraic closure.

Examples provide intuition. For instance, x^3 - 6x^2 + 11x - 6 factors as (x-1)(x-2)(x-3). A multivariate example is P(x,y) = x^2 y + 3xy^2 + 5.

Terminology and usage. The term polinomin is less common in modern English; most sources simply use polynomial. Variants exist in other languages where similar spellings denote the same concept.