Polynomialen

A polynomial, or polynomialen in some languages, is a mathematical expression formed by sums of constants and products of powers of variables, using only addition, subtraction, and multiplication with non‑negative integer exponents. In one variable x, a polynomial is typically written as P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where the leading coefficient a_n is not zero and the degree is n, the highest exponent that appears. Polynomials can also involve several variables, such as P(x, y) = a x^2 y + b x y^2 + c y^3, with the total degree determined by the exponents in each monomial.

Key concepts include monomials, coefficients, and the notion of degree. A polynomial with coefficients from a field or ring can be viewed as a function by evaluating it at numbers from that field. Roots (zeros) are values that make the polynomial equal to zero; over the complex numbers the Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots, counted with multiplicity.

Operations on polynomials include addition, subtraction, multiplication, and division with remainder. Factoring expresses a polynomial as a product of irreducible factors. Polynomials form algebraic structures known as polynomial rings, where the objects are polynomials and the operations mirror ordinary arithmetic.

Applications are wide, including solving equations, modeling and curve fitting, interpolation, and numerical methods. In computer science and numerical analysis, polynomials serve as building blocks for algorithms, with specialized techniques such as Horner’s method for efficient evaluation and FFT-based methods for fast multiplication. An example is P(x) = 2x^3 - 5x^2 + x - 7.