Master Polynomial Equations

Polynomial equations are equations of the form P(x) = 0, where P is a polynomial with coefficients in a field, often the field of rational numbers. Algebraic equations refer only to univariate equations, that is polynomial equations that involve only one variable. Polynomial equations may involve several variables, and the term polynomial equation is usually preferred to algebraic equation in the case of several variables. Some polynomial equations with rational coefficients can be solved algebraically, and a large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation and of the common solutions of several multivariate polynomial equations. The term algebraic equation dates from the time when the main problem of algebra was to solve univariate polynomial equations. Algebraic equations are the basis of a number of areas of modern mathematics such as algebraic number theory, Galois theory, field theory, transcendental number theory, Diophantine equation, and algebraic geometry. Two equations are equivalent if they have the same set of solutions, and the study of algebraic equations is equivalent to the study of polynomials. The fundamental theorem of algebra states that the field of the complex numbers is closed algebraically, and all polynomial equations with complex coefficients and degree at least one have a solution. A monic polynomial of odd degree must necessarily have a real root.Solving polynomial equations: techniques and methods

• Polynomials are algebraic expressions consisting of variables and coefficients.
• A polynomial equation is an equation in which two polynomials are set equal to each other.
• The degree of a polynomial is the highest power of the variable in the polynomial.
• The Fundamental Theorem of Algebra states that every polynomial equation of degree n has n roots.
• Abel showed that it is not possible to find a formula in general for equations of degree five or higher using only the four arithmetic operations and taking roots.
• Galois theory provides a criterion for determining whether the solution to a given polynomial equation can be expressed using radicals.
• If an equation has a rational root, the associated polynomial can be factored to give a new equation of lower degree.
• A common preliminary step in solving an equation of degree n is to eliminate the degree-n - 1 term.
• Cardano's formula is a well-known method for solving cubic equations.
• Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions.
• Polynomials of degree 5 are solvable using elliptical functions.
• Numerical approximations can be used to find roots of higher-degree equations.