Polynomial

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Calculus and Analysis

In calculus and analysis, a polynomial function, or polynomial for short, is a function of the form

f(x) = an xn + an-1 xn-1 + ... + a1 x + a0

where x is a variable ranging over the real (or complex) numbers and the a0, ... , an are fixed real (or complex) numbers called the coefficients of f. The highest occuring power of x is called the degree of f; its coefficient is called the leading coefficient. a0 is called the constant coefficient of f.

Polynomials of degree 0 are called constant functions, polynomials of degree 1 are called linear functions, polynomials of degree 2 are called quadratic functions and polynomials of degree 3 are called cubic functions. The function f(x) = -7 x3 + 2/3 x2 - 5 x + 3 is an example of a cubic function with leading coefficient -7.

Polynomials are important because they are the simplest functions: their definition involves only addition and multiplication (since the powers are just shorthands for repeated multiplications). They are also simple in a different sense: the polynomials of degree ≤ n are precisely those functions whose (n+1)st derivative is identical zero. One can view calculus as the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylors 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 rational functions. These are the only functions that can be evaluated directly on a computer, since typically only the operations of addition, multiplication and division 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 rational functions.

In order to determine function values of polynomials for given values of the variable x, one does not follow the proscription of the 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.

A root or zero of the polynomial f(x) is a number r such that f(r) = 0. 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 (non-constant) polynomial has a root (see Fundamental Theorem of Algebra).

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.

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 no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree ≥ 5 in terms of its coefficients (see [[Abel-Ruffini theorem]]). This result form the start of Galois theory which engages in a detailed study of relations among roots of polynomials.

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

f(x,y,z) = 2 x2 y z3 - 3 y2 + 5 y z - 2

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

In abstract algebra, one has to carefully distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form

f = an Xn + an-1 Xn-1 + ... + a1 X + a0

where the coefficients a0, ... , an 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
Xk Xl = Xk+l for all natural numbers k and l.

One can then check that the set of all polynomials with coefficients in the ring 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 R[X] as arising from R by adding one new element X to R and only requiring that X commute with all elements of R. In order for R[X] to form a ring, all sums of powers of X have to be included as well. Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the clean construction of the finite fields involves the use of those operations, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).

To every polynomial f in R[X], one can associate a polynomial function with domain and range equal to R. One obtains the value of this function for a given argument r by everywhere replacing the symbol X in f's expression by r. The reason that algebraists have to distinguish between polynomials and polynomial functions is that over some rings R (for instance over finite fields), two different polynomials may give rise to the same polynomial function. This is not the case over the real or complex numbers and therefore analysts don't separate the two concepts.

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], we say that f divides g if there exists a polynomial q in R[X] such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X - r) divides f. The converse is also true. The quotient can be computed using the Horner scheme.

If F is a field and f and g are polynomials in F[X] with g ≠ 0, then there exist polynomials q and r in F[X] with

f = q g + r

and such that that the degree of 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 "long division" and shows that the ring F[X] is a euclidean domain.

One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate 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.


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