Polinomiale

Polinomiale is the term used in Italian and Romanian to denote the mathematical object commonly called a polynomial in English. A polynomial in one or more variables is a finite sum of terms consisting of a coefficient multiplied by a nonnegative integer power of the variable(s). For a single variable x over a field F, a polynomial P is written P(x)=a_n x^n+...+a_1 x+a_0 with coefficients a_i in F and n≥0. The degree of P, when P is not the zero polynomial, is the largest index n with a_n ≠ 0; the zero polynomial is often assigned undefined or −∞ as its degree. A polynomial with leading coefficient a_n is called monic when a_n=1.

Polynomials form a ring, denoted F[x], under ordinary addition and multiplication; they can also be composed. They are closed under addition, subtraction, and multiplication; division with remainder is available, yielding the quotient and remainder by polynomial long division.

Over an algebraically closed field such as the complex numbers, every nonconstant polynomial P(x) has at least one root, and can be factored into linear factors: P(x)=a_n(x−r1)^{m1}…(x−rk)^{mk}. Over the real numbers, factors may be linear or irreducible quadratics. The Fundamental Theorem of Algebra ensures existence of complex roots; rational or integer roots are constrained by the rational root theorem.

Applications include interpolation (Lagrange, Newton), approximation (Taylor polynomials), numerical methods, control theory (characteristic polynomials), and coding theory. Efficient evaluation uses Horner’s method; factoring and root-finding are central problems in algebra and numerical analysis.