Solving Quintic Equations
Overview
By the nineteenth century, mathematicians had long been interested in solving equations called polynomials. However, Paolo Ruffini (1765-1822) and Niels Abel (1802-1829) proved that some polynomials could not be solved by previously known methods. Partly in response, Evariste Galois (1811-1832) developed a new way of analyzing and working with these types of equations. This method is called group theory, and it was to have implications in other scientific fields, such as mineralogy, physics, and chemistry.
Background
Polynomial equations are used in almost every branch of mathematics and science. An example of a polynomial equation is 3x2 + 4x + 5 = 0. This equation is called a second degree polynomial because the highest power of x it contains is 2. The degree of a polynomial indicates the number of solutions it has. A number is said to be a solution of a polynomial equation if substituting it into the equation makes the equation true. For instance, the number 7 is a solution of the equation x + 5 = 12. By the nineteenth century, mathematicians had already discovered ways to solve second, third, and fourth degree polynomial equations. They next turned their attention to solving fifth degree, or quintic, equations.
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