A polynomial is an expression built from a variable (or variables) and constants using only addition, subtraction, and multiplication — with the variable never appearing under division, inside a root, or as an exponent. Polynomials are among the most studied objects in mathematics, appearing from elementary algebra through to advanced algebraic geometry.
A polynomial is an expression like 3x² + 2x − 5, or just x + 1, or even the single number 7. The rule is simple: you can add, subtract, and multiply the variable by itself and by numbers, but you can never divide by the variable, put it under a square root, or raise it to a negative or fractional power.
Polynomials are the "nice" functions — they're smooth, continuous, and easy to compute with just addition and multiplication. Computers and calculators often approximate complicated functions (like sine or the exponential) using polynomials, because they're so much easier to calculate quickly.
where n is the degree (aₙ ≠ 0) and a₀, ..., aₙ are the coefficients, typically drawn from a field like ℝ or ℂ.
Every non-constant polynomial with complex coefficients has at least one complex root. By repeated factoring, a degree-n polynomial has exactly n roots when counted with multiplicity (a root can "repeat"). This was first rigorously proved by Gauss (1799), with his later proofs (1815, 1816) removing gaps in the original.
Every polynomial with real coefficients can be factored into linear and irreducible quadratic factors over ℝ, or entirely into linear factors over ℂ. For example: x² − 5x + 6 = (x−2)(x−3), with roots 2 and 3.
For ax² + bx + c = 0 (a ≠ 0):
This formula was known in essentially complete form to Indian mathematicians (Brahmagupta, 628 CE, and later Śrīdhara) and Babylonian mathematicians (geometric methods, c. 2000 BCE) long before its modern algebraic notation.
Formulas analogous to the quadratic formula exist for degree 3 (Cardano's formula, 1545) and degree 4 (Ferrari's method, 1540), but Galois theory proves no such general formula exists for degree 5 or higher.
The set of all polynomials with coefficients in a ring R, written R[x], is itself a ring under polynomial addition and multiplication. R[x] shares many properties with the integers: if R is a field, R[x] is a Euclidean domain, supporting polynomial division with remainder analogous to integer division.
Polynomials in several variables, like p(x,y) = x²y + 3xy² − 2, form the ring R[x₁,...,xₙ]. The set of points where a collection of multivariate polynomials simultaneously vanish is called an algebraic variety — the central object of study in algebraic geometry. Hilbert's Nullstellensatz (1893) establishes a precise correspondence between algebraic varieties and ideals in polynomial rings.
A polynomial in several variables is symmetric if it's unchanged by any permutation of its variables. The Fundamental Theorem of Symmetric Polynomials states every symmetric polynomial can be written uniquely as a polynomial in the elementary symmetric polynomials. This connects to Galois theory: the coefficients of a polynomial are symmetric functions of its roots, which is why the Galois group (acting on the roots) preserves the coefficients.
The discriminant of a polynomial encodes information about repeated roots: it is zero exactly when the polynomial has a repeated root. For ax²+bx+c, the discriminant is b²−4ac — this is why the sign of b²−4ac in the quadratic formula tells you whether the roots are real and distinct, real and repeated, or complex.
For systems of multivariate polynomial equations, Gröbner bases (Buchberger, 1965) provide an algorithmic method analogous to Gaussian elimination for linear systems, but for polynomial systems. This underlies modern computer algebra systems (Mathematica, Maple, SageMath) and has applications in robotics, coding theory, and cryptography.
Certain families of polynomials (Legendre, Chebyshev, Hermite, Laguerre) satisfy orthogonality relations with respect to specific inner products, making them essential tools in numerical analysis, quantum mechanics (Hermite polynomials appear in the quantum harmonic oscillator), and approximation theory.