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## Polynomial
In algebra, a
x is a scalar-valued variable, n is a nonnegative integer, and a_{0},...,a_{n} are fixed scalars, called the coefficients of the polynomial f. The highest occurring power of x (n if the coefficient a is not zero) is called the _{n}degree of f; its coefficient is called the leading coefficient. Where the leading coefficient is 1, we describe the polynomial as monic. a_{0} is called the constant coefficient of f. Each summand of the polynomial of the form a_{k} x^{k} is called a term.
The polynomial can be written in sigma notation as:
## Polynomials of low degree- 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*.
## Polynomials and calculus
Note that the polynomials of degree ≤ 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 Weierstrass approximation 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.
Quotients of polynomials are called ## Efficient evaluationIn order to determine function values of polynomials for given values of the variable x, one does not apply the polynomial as a formula directly, but uses the much more efficient Horner scheme instead. If the evaluation of a polynomial at many equidistant points is required, Newton's difference method reduces the amount of work dramatically. 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. ## Roots
A 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. These algorithms are studies in numerical analysis. ## Formulae for roots
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 sixteenth century (see quadratic formula, Cardano, Tartaglia). But formulas for degree 5 eluded researchers for a long time. In 1824, Abel proved the striking result that there can be ## Several variablesThetotal 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.## Complexity
In computer science, we say that a polynomial of highest order
^{4}). From the definition of order, |f(x)| ≤ C |g(x)| for all x>1, where C is a constant. Proof: - where x > 1
- because x
^{3}< x^{4}, and so on.
^{4})## Abstract algebra
In abstract algebra, one must take care to distinguish between
A
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
*X**a*=*a**X*for all elements*a*of the ring*R**X*^{k}*X*^{l}=*X*^{k+l}for all natural numbers*k*and*l*.
R forms itself a ring, the ring of polynomials over R,
which is denoted by R[X]. If R is commutative, then R[X] is an
algebra over R.
One can think of the ring
To every polynomial ## Divisibility
In commutative algebra, one major focus of study is
If *f*=*q**g*+*r*
r is smaller than the degree of g. The polynomials q and r are uniquely determined by f and g. This is called "division with remainder" or "polynomial long division" and shows that the ring F[X] is a Euclidean domain.Analogously we can define polynomial "primes" (more correctly, irreducible polynomials) which cannot be factorized into the product of two polynomials of lesser degree. Depending on the degree of the polynomial to be considered, simply checking if the polynomial has linear factors can eliminate several cases, and then resorting to checking divisibility of some other irreducible polynomials, however Eisenstein's criterion can be used to more efficiently determine irreducibility. ## More variables
One also speaks of polynomials in several variables, obtained by
taking the ring of polynomials of a ring of polynomials: Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. Other related objects studied in abstract algebra are formal power series, which are like polynomials but may have infinite degree, and the rational functions, which are ratios of polynomials. ## Special polynomials- Polynomial sequence
- Chebyshev polynomials
- Ehrhart polynomial (It is appropriate that this title is singular although some of the other special polynomials named after persons that are listed here are plural, because those are special polynomial
*sequences*.) - Hermite polynomials
- Hurwitz polynomial (It is appropriate that this title is singular although some of the other special polynomials named after persons that are listed here are plural, because those are special polynomial
*sequences*.) - Legendre polynomials
- Polynomial interpolation
- Binomial type
- Sheffer sequence
- List of polynomial topics
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