A polynomial is a mathematical expression involving a sum of powers in one or more variables
multiplied by coefficients. A polynomial
in one variable (i.e., a univariate
polynomial) with constant coefficients
is given by
 |
(1)
|
The individual summands with the coefficients (usually) included are called monomials
(Becker and Weispfenning 1993, p. 191), whereas the products of the form  in the multivariate case, i.e., with the
coefficients omitted, are called terms
(Becker and Weispfenning 1993, p. 188). The highest power
in a univariate polynomial is called its order,
or sometimes its degree.
Any polynomial  with  can be expressed
as
 |
(2)
|
where the product runs over the roots  of  and it
is understood that multiple roots are counted with multiplicity.
A polynomial in two variables (i.e., a bivariate polynomial) with constant coefficients
is given by
 |
(3)
|
The sum of two polynomials is obtained by adding together the coefficients sharing the same powers of variables (i.e., the same terms)
so, for example,
 |
(4)
|
and has order less than (in the case of cancellation of leading terms) or equal to the maximum order of the original two polynomials. Similarly, the product of two polynomials is obtained by multiplying term by term and combining the results, for example
and has order equal to the sum of the orders of the two original polynomials.
A polynomial quotient
 |
(7)
|
of two polynomials  and  is known as
a rational function. The process
of performing such a division is called long
division, with synthetic
division being a simplified method of recording the division.
For any polynomial  ,  divides  , meaning that the polynomial quotient is
a rational polynomial or,
in the case of an integer polynomial,
another integer polynomial (pers. comm., N. Sato, Nov. 23, 2004).
Exchanging the coefficients of a univariate polynomial
end-to-end produces a polynomial
 |
(8)
|
whose roots are reciprocals  of the original
roots  .
Horner's rule provides a computationally efficient method of forming a polynomial from a list of its coefficients, and can
be implemented in Mathematica
as follows.
Polynomial[l_List, x_] := Fold[x #1 + #2&, 0, l]
The following table gives special names given to polynomials of low orders.
Polynomials of fourth degree may be computed using three multiplications and five
additions if a few quantities are calculated first (Press et al. 1989):
![a_0+a_1x+a_2x^2+a_3x^3+a_4x^4=[(Ax+B)^2+Ax+C][(Ax+B)^2+D]+E,](/images/equations/Polynomial/NumberedEquation7.gif) |
(9)
|
where
Similarly, a polynomial of fifth degree may be computed with four multiplications and five additions, and a polynomial of sixth degree may be computed with four multiplications and seven additions.
Polynomials of orders one to four are solvable using only rational operations and finite root extractions. A first-order
equation is trivially solvable. A second-order equation is soluble using the quadratic equation. A third-order equation is solvable using
the cubic equation. A fourth-order
equation is solvable using the quartic
equation. It was proved by Abel and Galois using group theory that general equations of fifth and higher order
cannot be solved rationally with finite root
extractions (Abel's
impossibility theorem).
However, solutions of the general quintic equation may be given in terms of Jacobi
theta functions or hypergeometric
functions in one variable. Hermite and Kronecker proved that higher order polynomials
are not soluble in the same manner. Klein showed that the work of Hermite was implicit
in the group properties of the icosahedron. Klein's method of solving the quintic in terms
of hypergeometric functions
in one variable can be extended to the sextic, but for higher order polynomials,
either hypergeometric functions
in several variables or "Siegel functions" must be used (Belardinelli 1960,
King 1996, Chow 1999). In the 1880s, Poincaré created functions which give
the solution to the  th order polynomial equation in finite
form. These functions turned out to be "natural" generalizations of the
elliptic functions.
Barbeau, E. J. Polynomials. New York: Springer-Verlag, 1989.
Belardinelli, G. "Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébriques générales."
Mémoral des Sci. Math. 145, 1960.
Bini, D. and Pan, V. Y. Polynomial and Matrix Computations, Vol. 1: Fundamental Algorithms.
Boston, MA: Birkhäuser, 1994.
Borwein, P. and Erdélyi, T. Polynomials and Polynomial Inequalities. New York: Springer-Verlag,
1995.
Chow, T. Y. "What is a Closed-Form Number." Amer. Math. Monthly 106,
440-448, 1999.
Cockle, J. "Notes on the Higher Algebra." Quart. J. Pure Applied Math. 4,
49-57, 1861.
Cockle, J. "Notes on the Higher Algebra (Continued)." Quart. J. Pure
Applied Math. 5, 1-17, 1862.
King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser,
1996.
Mignotte, M. and Stefănescu, D. Polynomials: An Algorithmic Approach. Singapore: Springer-Verlag,
1999.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press, 1989.
Project Mathematics. "Polynomials." Videotape. http://www.projectmathematics.com/polynom.htm.
Ram, R. "Sums of Powers." http://users.tellurian.net/hsejar/maths/sumsofpowers/.
Weisstein, E. W. "Books about Polynomials." http://www.ericweisstein.com/encyclopedias/books/Polynomials.html.
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168,
2002.
|
Contact the MathWorld Team
