Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.
HISTORY OF NOTATION:
§ The history of algebra began in ancient Egypt and Babylon,where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved.
§ Later, Alexandrian mathematicians, the hero of Alexandria andDiophantus, took the ideas that the Egyptians and Babylonians had come up with and expanded upon them. Their knowledge became a staple of Islamic world, where it became known as "the science of restoration and balancing."
§ The Arabic word for restoration, al-jabru, became the root for the word algebra. In the 9th century, the Arab mathematicianal-Kwharizmi wrote one of the first books on Arabic algebra, and it provided examples and proofs of what we now know to be basic algebraic theory.
§ By the end of the 9th century, another Arab mathematician,
Abu Kamil, had expanded even
further on al-Kwharizmi's theories and was
able to prove the basic laws and identities of algebra and solve more
§ In about 300 BC, Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation. Euclid had no notion of equation, coefficients etc. but worked with purely geometrical quantities.
§ Hindu mathematicians had taken the Babylonian methods so further that, Brahmagupta (598-665 AD) gave an, almost modern, method which admits negative quantities. He also used abbreviations for the unknown, usually the initial letter of a colour was used, and sometimes several different unknowns occur in a single problem.
§ ByMidevil times Islamic mathematicians were able to discuss the importance of the unknown variable x. They were able to multiply, divide, and find the roots of polynomials and they started to put together binomial theorems.The Persian mathematician Omar Khayyamshowed how to find the roots of cubic equations through line segments of intersected conic sections, but was unable to come up with an equation for cubic polynomials.
§ In the early 13th century, however, Leonardo Fibonacci achieved a close approximation of the cubic equation: x3+2x2+cx=d.
§ In the early 16th century, Italian mathematiciansScipone del Ferro, NiccolóTartaglia, and GerolamoCordanowere able to solve the general cubic equation in terms of the constants in front of the variables.
§ Ludovico Ferrari found exact equations for polynomials up to the fourth degree.
§ René Decartes, an extremely important mathematician from the 16th century, discovered analytic geometry, which reduces the solutions of geometric problems into solutions in terms of algebraic equations. He also made significant contributions to the theory of equations, including coming up with what he called "the rule of signs" for finding the positive and negative roots of equations. In La géometrie, 1637, he introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen here (ax2 + bx = c)
in the general formula for a polynomial, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.