polinomske

Polinomske are algebraic expressions built from a finite sum of terms of the form a_i x^i, where i is a nonnegative integer and the coefficients a_i come from a specified algebraic set such as the real or complex numbers, or more generally a ring or field. The highest exponent with a nonzero coefficient is called the degree of the polynomial. A polynomial with coefficients a_0, a_1, ..., a_n is often written p(x) = a_0 + a_1 x + ... + a_n x^n, with a_n ≠ 0. When the coefficients are drawn from a field, polynomials form a ring under the usual addition and multiplication, and when the leading coefficient is 1 they are called monic.

Polynomial functions arise by evaluating p at a number x, producing a real- or complex-valued function p: F → F. Operations on polynomials include addition, subtraction, multiplication, and, over a field, division with remainder, yielding the Euclidean algorithm and factorization into irreducible polynomials.

A central concept is the root of a polynomial, a number r such that p(r) = 0. Over the complex numbers, the Fundamental Theorem of Algebra states every nonconstant polynomial factors completely into linear factors p(x) = a_n (x - r_1)...(x - r_n). Real polynomials may have real roots or complex conjugate pairs; factorization over the rationals is constrained by the rational root test and irreducibility criteria.

Historically, polynomials have played a key role in algebra and calculus, with early work by Viète and Newton contributing to the terminology and theory. Polinomske are foundational in areas ranging from algebraic geometry to numerical methods.