For early occurences of geometric problems that lead today to quadratic equations see The origin of quadratic equation in actual practice. The equations proper, let alone polynomials themselves, do appear only relatively late. The first breakthrough was made by Diophantus in Arithmetica c. So for the first time it became possible to write down polynomials, albeit only up to degree six.
He even had symbols for constant terms and reciprocals of powers. Diophantus was also first to convert text problems into polynomial equations, and use some rudimentary verbal algebra to solve them. However, Diophantus's "polynomials" are not yet separate items either, they only appear as sides in equations. This is analogous to Babylonian and Hellenistic use of zero symbol as a placeholder.
The next great step was made by medieval Islamic mathematicians. The familiar rules of algebra were formalized by Al-Khowarizmi ca. Perhaps the first person to conceive of general polynomials is Al-Karaji AD , who can be called their "discoverer". He realized that the series of powers and their reciprocals extends indefinitely.
Correspondingly, he extended Diophantus's notation to write polynomials of arbitrary degree, and gave rules for their addition, subtraction and multiplication. He dispensed with verbal and semi-verbal notation, and used tables of coefficients to write and perform calculations with polynomials, including those with negative powers Laurent polynomials.
This greatly simplified all algebraic calculations with them because "laws of exponents" are applied automatically. And he gives the first polynomial division algorithm, the grandfather of modern long and synthetic division. This was the first mathematical justification of a positional division algorithm. Unfortunately, Renaissence Europe did not absorb Al-Samawal's innovations, and instead proceeded through incremental improvement of notation.
In particular, the notation used by del Ferro, Tartaglia and even Cardano to solve the cubic in s was largely verbal, and inferior to Al-Samawal. Even so, in Ars Magna Cardano introduced the technique of substitutions that not only solved the cubic and quartic, but became indispensable in polynomial algebra later.
See Why is "Cardano's Formula" wrongly attributed to him? In particular the use of letters for parameters allowed general consideration of polynomials rather than example by example. He connected it to the classical method of analysis and synthesis described by Pappus, but with conversion to algebraic equations in the middle. This not only streamlined solutions of many classical problems, but also covered many new ones, expressible by higher order equations.
Moreover, Descartes considers polynomials in two variables, which represented algebraic curves in analytic geometry, and this is where algebraic geometry takes its root. Descartes's formalization of construction methods, and classification of problems based on their algebraic representation, led to techniques required for impossibility proofs, starting with Gregory's unsuccessful attempt to prove algebraic unsolvability of quadrature The first nontrivial polynomial to be studied was the quadratic polynomial which is encountered very commonly.
Even so, its solutions in the most general case were fully understood only in the 16 th - 17th century, though essentially the quadratic formula was known by the 9 th century. Around the same time, mathematicians were also working on the cubic equation. The first solution to the general case was essentially solved by Tartaglia whose work has been reportedly plagiarized by Gerolamo Cardano.
However, the method has been historically referred to as the Tartaglia-Cardano method. There were some kinks in the method that had to be ironed out. The work on solving them, even though started with the Babylonians around BC, it culminated into its form that we now know, only in the 16 th century. The significant results in mathematics have never occurred over a fortnight. They always seem to be an improvement over the work till any given moment in time.
Naturally, mathematicians were curious about the next step, the quintic equation! Multiplying polynomials was invented in the 15'' century before that, equations were written out in words. Rene Descartes, is one of the person who introduced the concept of the graph of a polynomial equations, in La geometric on Polynomials are one f the most important concepts in algebra and throughout mathematics and science.
In mathematics, a polynomial is an expression construct from variables also known as Indeterminate and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents.
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