Polynomial equation

In mathematics, and more specifically algebra, a polynomial equation is an equation involving only polynomials, and where one is interested in finding solutions. Polynomial equations occur frequently in applications -- typically linear or quadratic, but but higher order equations do occur. For example, fifth order equations occur when deriving the existence of the five Lagrange points in the vicinity of two orbiting masses.

The most common polynomial equations in applications involve polynomials with real number coefficients. In this article, it is assumed that all polynomials have real number coefficients. See the "Advanced" subpage for information about polynomial equations with other types of coefficients.

Every polynomial equation in one variable involving real coefficients has finitely many solutions, a consequence of the fundamental theorem of algebra. This is not true for other types of equations. For instance,


 * $$ sin ( \pi x) = 0 $$

has an infinite solution set, namely, the set of integers.

The higher the order of a polynomial equation, the harder it is to solve the equation. Linear equations can be solved using only arithmetic operations. Quadratic equations can be solved using the quadratic formula. Cubic and quartic equations can also be solved using similar but much more complicated formulas, the cubic formula and quartic formula. Not only are these formulas more complicated, but applying them requires sophisticated knowledge of arithmetic with complex numbers. There are no analogous formulas for solving polynomial equations of order higher than four. For such equations, one often must resort to more sophisticated methods for find solutions. For many applications, only approximate solutions are needed. These can be found for any polynomial equation using techniques from calculus.

Polynomial equations involving several variables also occur in applications. For instance, the graph of a second degree polynomial equation in two variables is a conic section, which can describe the path of motion of an object moving under gravity or of a charged particle deflecting away from an oppositely charged one. Elliptic curves, which can be described using cubic polynomial equations, have important applications to cryptography. The study of solutions of general systems of polynomial equations in several variables is very difficult and was the genesis of the modern field of algebraic geometry, which itself is used in modern theoretical physics.

Solving polynomial equations
As mentioned above, quadratic, cubic, and quartic equations can be solved using the quadratic, cubic, and quartic formulas. There is no similar formula for general polynomial equations of fifth or higher orders. In fact, the quadratic, cubic, and quartic formulas show that roots of quadratic, cubic, or quartic equations can be found from the coefficients in the equation using the arithmetic operations and taking second, third, and fourth roots. Such equations are said to be solvable by radicals. By contrast, an arbitrarily chosen polynomial equation of order at least five will not be solvable by radicals, and in particular, no general formula specifying the solutions in terms of only the coefficients, arithmetic operations, and radicals can exist. The proof of this was one impetus for the development of Galois theory.

Some polynomial equations of order at least five can be solved by radicals, and ad hoc techniques (such as factoring or using the rational root theorem) can sometimes be successful for solving such equations. Techniques have been developed for solving other types of equations, such as expressing the solutions of a quintic equation in terms of arithmetic operations, radicals, and the root of a specific quintic polynomial. Other techniques involve introducing sophisticated functions and using calculus. But for many applications, exact solutions are not required, and in those cases the simplest approach is to find approximate solutions using ideas from calculus, such as Newton's method.