# Polynomial

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/polynomial/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
An expression built from a variable and constants using addition, subtraction, and multiplication only (no division by the variable).

## General form
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

## Fundamental Theorem of Algebra
Every non-constant complex polynomial has a complex root (Gauss, 1799, refined 1815/1816). Degree-n polynomial has exactly n roots (with multiplicity).

## Quadratic formula
x = (−b±√(b²−4ac))/2a. Known essentially in complete form to Babylonian (geometric, c. 2000 BCE) and Indian mathematicians (Brahmagupta 628 CE, Śrīdhara) long before modern notation.

## Sources
### Tier 1
- Lang, S. (2002). *Algebra*. Revised 3rd ed. Springer.
- Cox, D., Little, J. and O'Shea, D. (2015). *Ideals, Varieties, and Algorithms*. 4th ed. Springer.
### Tier 2
- Plofker, K. (2009). *Mathematics in India*. Princeton University Press.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*