{"id":"polynomial","domain":"algebra","type":"Concept","title":"Polynomial","slug":"polynomial","url":"/mathematics/algebra/polynomial/","summary":"An expression built from a variable and constants using addition, subtraction, and multiplication only. Studied via roots, factoring, and the Fundamental Theorem of Algebra.","formal_definition":"p(x) = aₙxⁿ + ... + a₁x + a₀ with coefficients in a ring or field.","key_results":[{"result":"Fundamental Theorem of Algebra","statement":"Every non-constant complex polynomial has a complex root (Gauss, 1799)."},{"result":"Quadratic formula","statement":"x = (-b±√(b²-4ac))/2a for ax²+bx+c=0."}],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Lang, S. (2002). Algebra. Revised 3rd ed. Springer."},{"tier":1,"citation":"Cox, D., Little, J. and O'Shea, D. (2015). Ideals, Varieties, and Algorithms. 4th ed. Springer."},{"tier":2,"citation":"Plofker, K. (2009). Mathematics in India. Princeton University Press."}],"relationships":[{"type":"forms","target":"ring","label":"Ring (polynomial ring)"},{"type":"forms","target":"vector-space","label":"Vector Space"},{"type":"studied_via","target":"galois-theory","label":"Galois Theory"},{"type":"historical_rules_from","target":"brahmagupta","label":"Brahmagupta"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}