The earliest work on quadratics -- degree 2 polynomials -- goes back almost 3,000 years. Quadratic problems are mentioned in Babylonian cuneiform tablets from 1900 BC. Around 600 BC, the legendary Indian mathematician Brahmagupta had a general solution for all quadratic equations. General solutions to third- and fourth-degree equations were not developed until the 16th century.
Cubics -- degree 3 polynomials -- were solved for special cases before the 16th century, which turned out to be the golden age of polynomial factoring. Scipione del Ferro and Niccolò Tartaglia were the superstars of royal contests to solve polynomials. Finally, Ferro developed a generalized method to solve any cubic. In 1526, on his deathbed, he confided his secret solution to a student named Fior. Ferro recovered but was too weak to travel to competitions. When Fior bragged that he was Ferro's successor, Tartaglia realized that Ferro had divulged the general solution. Tartaglia then discovered that general solution himself, and went on to defeat the unscrupulous Fior in competitions.
Quartic equations -- degree 4 polynomials -- were widely regarded as unsolvable in a general way. As the Inquisition took hold in Spain during the late 15th century, mathematicians were burned at the stake for even working on quartic equations. The equations were associated with the fourth dimension and the devil. The general solution to quartic equations was finally developed by Lodovico Ferrari -- a student of Tartaglia. The general solutions to cubic and quartic equations were published in "Ars Magna" ("The Magnificent Art") in 1545 by Gerolamo Cardano, a friend of Leonardo Da VInci.
The search for a general solution to fifth-degree equations continued unsuccessfully for centuries. In 1824, a Norwegian mathematician named Niels Henrik Abel used group theory to prove that there could be no general algebraic formula for the solution of fifth-degree equations. In 1830, the French mathematician Evariste Galois showed that Abel's proof could be extended to any higher degree polynomial. These men are two of the most tragic figures in the history of mathematics. Abel lived in poverty, teaching high school mathematics in a small town in Norway. Two weeks after Abel died, a letter arrived offering him a prestigious job at the University of Berlin.Galois was killed in a duel over a lady of questionable repute when he was only 20 years old.