Polynomial
From Academic Kids

 Polynomial (adjective): Of, relating to, or consisting of more than two names or terms.
In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders.
Because of their simple structure, polynomials are very easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.
In linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors.
With the advent of computers, polynomials have been replaced by splines in many areas in numerical analysis. Splines are piecewise defined polynomials and provide more flexibility than ordinary polynomials when defining simple and smooth functions. They are used in spline interpolation and computer graphics.
Contents 
History
Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (nonconstant) polynomial has a root: this is the statement of the fundamental theorem of algebra.
There is a difference between approximating roots and finding concrete closed formulas for them. Formulas for the roots of polynomials of degree up to 4 have been known since the 16th century (see quadratic equation, Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers for a long time. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see AbelRuffini theorem). This result marked the start of Galois theory which engages in a detailed study of relations among roots of polynomials.
The difference engine of Charles Babbage was designed to create large tables of values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's difference method.
Definition
For given constants (i.e., numbers) a_{0}, …, a_{n} in some field (possibly but not limited to R or C) with a_{n} nonzero, a polynomial (function) of degree n is a function of the form
 <math>f(x) = a_0 + a_1 x + \cdots + a_{n  1} x^{n  1} + a_n x^n.<math>
More concisely, the polynomial can be written in sigma notation as
 <math>f(x) = \sum_{i = 0}^{n} a_{i} x^{i}.<math>
The constants a_{0}, …, a_{n} are called the coefficients of the polynomial. a_{0} is called the constant coefficient and a_{n} is called the leading coefficient. When the leading coefficient is 1, the polynomial is called monic or normed.
Each summand a_{i} x^{i} of the polynomial is called a term. A polynomial with one, two or three terms is called monomial, binomial or trinomial respectively.
Polynomial functions of
 degree 0 are called constant functions,
 degree 1 are called linear functions,
 degree 2 are called quadratic functions,
 degree 3 are called cubic functions,
 degree 4 are called quartic functions and
 degree 5 are called quintic functions.
Examples
Some examples of polynomials of low degree:
Missing image Polynomialdeg2.png 
Missing image Polynomialdeg3.png 
Missing image Polynomialdeg4.png 
Missing image Polynomialdeg5.png 
The function
 <math>f(x)= 7x^3 + \begin{matrix}\frac{2}{3}\end{matrix} x^2  5x + 3<math>
is an example of a cubic function with leading coefficient −7 and constant coefficient 3.
Notes
The polynomials up to degree n form a vector space of dimension n + 1, which is sometimes called <math>\Pi_n<math> or <math>K_n[x]<math> (where K indicates the field of coefficients, e.g. K=R or C). In this article polynomials are written using the (canonical) monomial basis (i.e. 1, x, x^{2}, …, x^{n}), but it should be mentioned that other bases exist, for example the Chebyshev polynomials, which may be preferable depending on the problem domain.
Roots
A root or zero of a polynomial f is a number ζ so that f(ζ) = 0. The fundamental theorem of algebra states that a polynomial of degree n over the complex numbers has exactly n complex roots (not necessarily distinct ones). Therefore a polynomial can be factorized as
 <math>f(x) = a_n(x\zeta_1)\cdots(x\zeta_{n})<math>
where each ζ_{i} is a root of the polynomial f.
The AbelRuffini theorem in algebra states that generally there is no closed formula to calculate the roots of a polynomial of degree 5 or higher. Closed formula means a formula constructed using only the coefficients of the polynomial and the operations of addition, multiplication and exponentiation (and their inverse operations).
Numerical analysis
Polynomials and calculus
One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the StoneWeierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials are also frequently used to interpolate functions.
Quotients of polynomials are called rational functions. Piecewise rationals are the only functions that can be evaluated directly on a computer, since typically only the operations of addition, multiplication, division and comparison are implemented in hardware. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be approximated in software by suitable piecewise rational functions.
Evaluation of polynomials
The fast and numerically stable evaluation of a polynomial for a given x is a very important topic in numerical analysis. Several different algorithms have been developed for this problem. Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x.
To evaluate a polynomial in monomial form one can use the Horner scheme. For a polynomial in Chebyshev form the Clenshaw algorithm can be used. If several equidistant x_{n} have to be calculated one would use Newton's difference method.
Finding roots
As there is no general closed formula to calculate the roots of a polynomial of degree 5 and higher, rootfinding algorithms are used in numerical analysis to approximate the roots. Approximations for the real roots of a given polynomial can be found using Newton's method, or more efficiently using Laguerre's method which employs complex arithmetic and can locate all complex roots.
Several variables
In multivariate calculus, polynomials in several variables play an important role. These are the simplest multivariate functions and can be defined using addition and multiplication alone. An example of a polynomial in the variables x, y, and z is
 <math>f(x, y, z) = 2 x^2 y z^3  3 y^2 + 5 y z  2. \,<math>
The total degree of such a multivariate polynomial can be gotten by adding the exponents of the variables in every term, and taking the maximum. The above polynomial f(x, y, z) has total degree 6.
Abstract algebra
 Main article: polynomial ring.
In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form
 <math>f = a_n X^n + a_{n  1} X^{n  1} + \cdots + a_1 X + a_0<math>
where the coefficients a_{0}, ..., a_{n} are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive law and the rules
 <math>
 <math>