polinom

Polinom, or polynomial, is an algebraic expression formed by summing powers of one or more variables with constant coefficients. The exponents are nonnegative integers, and the coefficients come from a given coefficient set such as integers, real numbers, or complex numbers. In one variable x, a polynomial of degree n is written as a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, with a_n ≠ 0. The leading coefficient a_n determines the degree. The special case where all coefficients are zero is called the zero polynomial.

The degree of a polynomial is the largest exponent with a nonzero coefficient. If the polynomial involves several variables, its degree is the maximum total degree of its monomials. Polynomials in several variables can be written as sums of monomials like a_{i,j,...} x^i y^j ...

Polynomials form an algebraic structure that is closed under addition, subtraction, and multiplication. Division by a polynomial is not always exact, but division with remainder is possible (polynomial long division). When the coefficients come from a field, the set of polynomials in one variable forms a Euclidean domain and a ring; in multiple variables they form a polynomial ring R[x1,..., xn].

Roots and values: A value a is a root of a polynomial p if p(a) = 0. Over the complex numbers, a degree n polynomial has exactly n roots counted with multiplicity (Fundamental Theorem of Algebra).

Applications and examples: Polynomials model many phenomena, define polynomial functions, and serve in interpolation, approximation, and numerical methods. Examples include p(x) = 3x^2 + 2x − 5 and q(x, y) = x^2 + y^2.