The history of polynomials starts with very basic concepts of mathematics that many countries developed over time.
Among the mathematicians that have been documented, there are many famous Arabs, Hindus, Islams, Chinese, Egyptians,
and European thinkers that have furthered the progress of mathematics throughout history. Of course, people all over the world
were starting to develop more advanced mathematics at this time so it is difficult to say which part of the world was more mathematically
advanced during what period.
The earliest preserved mathematical expression resembling polynomials was found in China in 200 BCE.
(Arriaga, Jorge) This equation was simply an expression of the prices of three different qualities and quantities
of sheaves of grain that amounted to the price of 29 dou for all of them. In our standard mathematical writing today this
could be expressed as 3x+2y+z=29. (Ali Syed, Imran)
al-Kwharizimi on the left, Abu Kamil on the right.
Before polynomial equations, linear equations were being realized in Egypt and Babylon. They even discovered how to solve
quadratic equations in essentially the same methods that we use today. (History of Algebra) Later three Arabic men,
including al-Kwharizimi and Abu Kamil, expounded upon the works of the Egyptians and Babylonians and verified basic laws and
identities of algebra and established proofs. (Ali Syed, Imran) Over time, negative quantities were introduced into the past
methods, and x was also starting to be used as a variable with multiplication, division, and roots.
Left: Leonardo Fibonacci, Middle: Ludovico Ferrari, Right: Rene Descartes.
Eventually the cubic equation came about through Leonardo Fibonacci along with the methods for finding Pythagorean triples.
(O’Connor & Robertson, 1998) Following Fibonacci’s discoveries, three Italian mathematicians solved the general cubic equation
early in the 16th century. The fourth degree equations were then solved by Ludovico Ferrari. This was followed by Robert Recorde’s
use of the plus, minus, and equal signs in equations. (Ali Syed, Imran) One of the most important mathematicians, Rene Descartes
took over the mathematical discovery for a time and not only contributed greatly to the foundations of analytical geometry,
but also assisted “other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate
calculus, group theory and more.” (Anirudh, 2018) Descartes also standardized algebraic notation with a, b, and c being used for
known quantities, and x, y, and z being used for unknown quantities.
Adrien Marie Legendre on the left, Carl Friedrich Gauss on the right.
In regards to polynomial factorization, Adrien Marie Legendre was one of the first to invent a truly groundbreaking concept that
we still use today. It involves root problems and replacing x for x+a. The continual splitting of the partial factorization of x+a
is a simplification method that is used as the most effective solving strategy for probabilistic algorithms even today.
(Gathen, 2006) Carl Friedrich Gauss was another huge contributor to the factorization of polynomials. He discovered mass amounts
of earth shattering mathematical concepts including distinct-degree factorization (which means that Gauss discovered an algorithm that
splits a square-free polynomial into a product of polynomials whose irreducible factors all have the same degree.(Factorization of polynomials
over Finite fields)) Gauss’ discovery was then literally introduced straight into computer programs when it was discovered and translated out of
Latin. In fact, many of his other undiscovered ideas were thought of by future mathematicians and were named after his predecessors even though
Gauß was truly the first thinker on the subject. (Gathen, 2006)
