History of Polynomials

Polynomials began with the practical need to solve equations and, over more than four millennia, became the foundation of modern algebra. The history below traces how solution methods were discovered degree by degree.

1. Degree 1 — The Origin of Equations

The simplest polynomial equation \( ax + b = 0 \) is as old as civilization. The Rhind Papyrus of ancient Egypt (circa 1650 BCE) records linear equations such as "a quantity, plus its 1/7, becomes 19" (\( x + \frac{x}{7} = 19 \)). Babylonians routinely solved linear equations for grain distribution and labor accounting.

The solution \( x = -b/a \) looks trivial, but it presupposes the concepts of division and negative numbers. Although Chinese mathematics (the Nine Chapters on the Mathematical Art) and Indian mathematics used negative numbers far earlier, in Western mathematics they were not widely accepted until around the 17th century.

2. Degree 2 — Babylonian Completing the Square

Babylonian clay tablets from around 2000 BCE record algorithmic procedures for solving quadratics. A problem of the form "given the area of a square plus a multiple of its side, find the side" — in modern notation \( x^2 + bx = c \) — was solved essentially by completing the square.

The Greeks, notably Euclid in Elements, treated quadratic problems as geometric manipulations of line segments (application of areas). The algebraic notion of "polynomial" did not yet exist; everything was geometric.

In the 9th century, al-Khwārizmī's al-Jabr wa'l-Muqābala classified quadratics into six standard forms and gave systematic solutions for each. The book's name is the origin of the word algebra. Because negative and zero coefficients were not accepted, the cases \( x^2 = bx \), \( x^2 + bx = c \), and \( x^2 + c = bx \) were treated as distinct types.

3. Degree 3 — The Renaissance Equation Wars

In the 11th century the Persian poet-mathematician Omar Khayyam classified cubics into 19 types and solved them geometrically as intersections of conic sections, but obtained no algebraic formula.

The breakthrough came from 16th-century Italian mathematicians. Around 1515 Scipione del Ferro of Bologna found a formula for \( x^3 + px = q \) and kept it secret. Tartaglia independently rediscovered it, and Cardano published it in his 1545 Ars Magna. The resulting Cardano formula reads

The formula sometimes routes through the square roots of negative numbers even when the roots are real (the casus irreducibilis). This triggered the development of complex numbers.

4. Degree 4 — Ferrari's Solution

Cardano's student Ferrari reduced the quartic equation to an auxiliary cubic (the resolvent) and thereby solved it. The method also appears in Ars Magna.

With this, every polynomial equation of degree 1 through 4 was known to be solvable in radicals (the four arithmetic operations and \( n \)-th roots).

5. Development of Symbolic Algebra

Alongside the solution formulas, the very language for polynomials evolved.

• Diophantus (3rd century): the first to systematically use shorthand symbols for equations.
• François Viète (late 16th century): established symbolic algebra using letters and discovered the relations between roots and coefficients (Vieta's formulas).
• Descartes (17th century): connected polynomials with curves via coordinate geometry, and gave the rule of signs bounding the number of positive real roots.

Viète's contribution shifted perspective from solving particular equations to discussing properties of roots in general.

6. Degree 5 and Above — Proving the Unsolvability of the General Quintic

In the 18th century Lagrange analyzed the degree-3 and -4 solutions in terms of permutations and suggested that the same method could not solve quintics. This focus on permutations of the roots became the seed of group theory.

In 1799 Gauss gave the first substantial proof of the fundamental theorem of algebra (a fully rigorous version came later, with Argand and Cauchy): a degree-\( n \) polynomial over \( \mathbb{C} \) has \( n \) roots. Roots exist; whether they can be written as a formula is another question.

In 1824 Abel proved that "the general quintic cannot be solved by radicals" (the Abel–Ruffini theorem). Abel himself died young, at the age of 26.

In 1832 Galois completely characterized solvability via the structure of groups. He created the theory (Galois theory) that decides whether a given polynomial is solvable by radicals. The fundamental reason the general quintic has no radical formula is that the symmetric group \( S_5 \) is not solvable. Galois died in a duel at just 20; the manuscript he wrote the night before became the starting point of today's Galois theory.

7. Timeline

8. References

• B. L. van der Waerden, A History of Algebra: From al-Khwārizmī to Emmy Noether, Springer, 1985.
• J. Stillwell, Mathematics and Its History, 3rd ed., Springer, 2010.
• V. J. Katz, A History of Mathematics, 3rd ed., Addison-Wesley, 2008.
• Wikipedia: History of algebra, Galois theory.