A polynomial is an algebraic expression in which whole-numbered powers of the variable multiplied by numerical coefficients are added together. The powers of the variable must be positive integers, or zero. An example of a polynomial is 3x3 + 7x - 2. A polynomial equation has a polynomial expression, or zero, on each side of the equals sign: for example, 4a2 - 5.6a + 1.7 = 0 or 5x2 - 1 = 8x + 2x4. The type of polynomial expression or equation is determined by the number of terms present and by the highest power present of the variable. A binomial is a polynomial of two terms, a trinomial is a polynomial of three terms. A quadratic, for example, has nothing higher than a squared term. Cubics, quartics, and quintics have maximum powers of three, four, and five, respectively.
Mathematicians who have done pioneering work on each of these higher types of polynomial equations have, for some reason, tended to have had colorful and star-crossed lives. Niccolo Tartaglia, who first solved the cubic, failed miserably for the rest of his life, largely because he spent it trying to discredit Girolamo Cardano. Tartaglia told Cardano his method of solution and swore him to secrecy but Cardano went ahead and published the solution anyway. Cardano himself lived a long unhappy life and his only son was executed for murder. Lodovico Ferrara, Cardano's student, who solved the general quartic, was poisoned, probably by his sister, over an inheritance dispute. Finally, Evariste Galois, who showed the general quintic was unsolvable, died in a duel at the age of 20.