Polinomial

A polynomial is an expression built from variables and coefficients using addition, subtraction, and multiplication, with nonnegative integer exponents. In one variable x, a polynomial can be written in standard form as a0 + a1 x + a2 x^2 + ... + an x^n, where an ≠ 0. The ai are coefficients from a given field or ring, and n is the degree of the polynomial.

Notation and structure: The monomials ai x^i are the terms; the leading term is the one with the highest exponent, and the leading coefficient is an. Polynomials can have multiple variables, for example P(x, y) = 3x^2 + 2xy + y^2.

Operations: Polynomials are closed under addition, subtraction, and multiplication. Division with remainder yields a quotient Q and a remainder R such that P = QD + R, with deg R < deg D. Over a field, the division algorithm holds. Factorization into irreducibles plays a central role in understanding roots and structure, and a polynomial splits into linear factors when it has a full set of roots in the chosen field.

Roots and factorization: A polynomial of degree n has up to n roots in the complex numbers, counting multiplicities. Real roots correspond to real linear factors; the Rational Root Theorem gives possible rational zeros. Vieta’s formulas relate sums and products of the roots to coefficients.

Applications and examples: Polynomials model a wide range of phenomena, from simple growth to curves used in interpolation and approximation. Polynomials with coefficients in a ring or field form a polynomial ring, a fundamental object in algebra. Common examples include x^2 + 3x + 2 and x^3 − 4x.