Monomials, binomials and trinomials are special cases of polynomials with one, two and three terms respectively.
The polynomial can be written in sigma notation as:
- 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.
Note that the polynomials of degree ≤ n are precisely those functions whose (n+1)st derivative is identically zero.
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 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.
In 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.
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 marked 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
- where x > 1
- because x3 < x4, and so on.
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
- 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 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 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
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.
- 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