[Math Lair] Theory of Equations

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Theory of equations is a branch of algebra that investigates properties of polynomial equations. Questions that the theory of equations investigates include:

A polynomial equation of degree n is an equation of the form

a0 + a1x + a2x² + ... + anxn = 0
Where an ≠ 0 (otherwise it would be an equation of degree n − 1) and where all of the a's are integers (whether positive or negative).

To take an example, 3 − 4x + x² = 0 is a polynomial equation of degree 2.

An equation of degree 1 is called a linear equation, one of degree 2 a quadratic equation, one of degree 3 a cubic equation, one of degree 4 a quartic equation, one of degree 5 a quintic equation, and so on.

As for whether a polynomial equation has a solution and how many, the fundamental theorem of algebra states that a polynomial equation of degree n must have n roots, although some roots may be repeated and/or complex. This was not proved until the 19th century, although it had been suspected for centuries before.

Investigations into the theory of equations, particularly how to solve polynomial equations, have taken place as far back as written records exist. The earliest known mathematical writings contain polynomial equations. The Rhind Papyrus, from ancient Egypt, contains solutions of linear equations. Babylonians solved certain types of quadratic equations. The Indians and Arabs would later find the general solution to the quadratic equation. These two cultures would also investigate cubic equations, although the general solution to the cubic would not be found until the sixteenth century. Girolamo Cardano first published the solution, based in part on the work of other Italian mathematicians. Cardano also published the solution to the quartic equation.

It was discovered in 1824 that there is no formula for solving the general quintic equation. While there are ways to solve some quintic equations, consider the equation x5x + 1. It has been proven that the solution to this equation cannot be found by the normal operations used to solve polynomial equations: addition, subtraction, division, and taking roots (square roots, cube roots, etc.), and the solution to this equation cannot be expressed in radicals. The solutions to quintic equations require what are called Bring radicals, also known as ultraradicals or hyperradicals. There are also methods of finding approximations to the solutions to any degree of accuracy that suffice for most practical purposes.

Sources used (see bibliography page for titles corresponding to numbers): 48.