polynoomi

Polynoomi, or polynomial, is a mathematical expression built from a finite sum of terms, each consisting of a constant coefficient multiplied by a nonnegative integer power of a variable. The standard form in one variable x is a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n ≠ 0. The highest power n is the degree of the polynomial; the coefficient a_n is the leading coefficient. The zero polynomial has all coefficients zero and is a special case with undefined degree.

Polynomials can involve more than one variable (multivariate polynomials) and can have real, complex, or other coefficient types. They are closed under addition and multiplication, and their variables can be substituted with numbers to yield numeric values (polynomial functions).

Key concepts include factoring and roots. A root or zero is a value that makes the polynomial equal to zero. Over the complex numbers, a nonconstant polynomial of degree n has exactly n roots counting multiplicities (Fundamental Theorem of Algebra). Factoring into polynomials with smaller degree reveals those roots; over the rationals or integers, various theorems help identify rational roots.

Polynomials form a ring under standard addition and multiplication, and they are central to numerical methods, algebra, and modeling. Applications include interpolation and curve fitting, solving equations, control theory, physics, and computer algebra. The term polynomial derives from Latin poly- “many” and nomial “term” (or from Greek poly- and nomos).