polynomiaaleihin

Polynomiaali (Finnish for polynomial) is a mathematical expression formed by a sum of terms in which a variable appears with nonnegative integer powers and coefficients multiply those powers. In one variable, a polynomial is written as p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a nonnegative integer and the a_i are coefficients. Polynomials can also involve several variables, in which case they are called multivariate polynomials, for example p(x, y) = a x^2 + b xy + c y^2 + d x + e y + f.

The coefficients of a polynomial may come from any field or ring; common choices are real numbers or complex numbers. Polynomials over a field form a algebraic structure known as a polynomial ring. The variable is often denoted x (or several variables in the multivariate case), and the degree of a polynomial is the highest exponent with a nonzero coefficient. The zero polynomial is a special case with no defined degree.

Operations on polynomials include addition, subtraction, multiplication, and division with remainder. Division yields a quotient and a remainder, and polynomials can often be factored into irreducible components. Roots (or zeros) are values of the variable that make the polynomial equal to zero, and they are central to factorization.

Key results include the Fundamental Theorem of Algebra, which asserts that a non-constant polynomial with complex coefficients has as many roots as its degree (counting multiplicities). Over the real numbers, roots are either real or occur as complex conjugate pairs. Polynomials have wide applications in solving equations, interpolation, approximation (Taylor polynomials), and numerical methods, and they extend to multiple variables in areas such as algebraic geometry and multivariate analysis.