User Contributed Dictionary
Pronunciation
 /pɑ.lə.noʊ.mi.l/
Noun
polynomials Plural of polynomial
Extensive Definition
In mathematics, a polynomial is
an expression
constructed from one or more variables and constants, using the operations
of addition, subtraction, multiplication, and constant positive
whole number exponents. For example, x^2  4x + 7\, is a
polynomial, but x^2  4/x + 7x^\, is not because it involves
division by a variable and has an exponent that is not a positive
whole number.
Polynomials are one of the most important
concepts in algebra and throughout mathematics and science. They
are used to form polynomial equations, which encode a wide range of
problems, from elementary
story problems to complicated problems in the sciences; they
are used to define polynomial functions, which appear in settings
ranging from basic chemistry and physics to economics, and are used in
calculus and numerical
analysis to approximate other functions. Polynomials are used
to construct polynomial
rings, one of the most powerful concepts in algebra and algebraic
geometry.
Overview
A polynomial is either zero, or can be written as the sum of one or more nonzero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) multiplied by zero or more variables (which are usually represented by letters). Each variable may have an exponent which is a nonnegative integer. The exponent on a variable in a term is equal to the degree of that variable in that term. Since x=x^1, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a constant term is 0. The coefficient of a term may be any number, including fractions, irrational numbers, negative numbers, and complex numbers.For example,
 5x^2y\,
is a term. The coefficient is –5, the
variables are x and y, the degree of x is two, and the degree of y
is one.
The degree of the entire term is the sum of the
degrees of each variable in it. In the example above, the degree is
2 + 1 = 3.
A polynomial is a sum of terms. For example, the
following is a polynomial:
 3x^2  5x + 4\,.
It consists of three terms: the first is degree
two, the second is degree one, and the third is degree zero. Here
"" stands for "", so the coefficient of the middle term
is −5.
When a polynomial in one variable is arranged in
the traditional order, the terms of higher degree come before the
terms of lower degree. In the first term above, the coefficient
is 3, the variable is x, and the exponent
is 2. In the second term, the coefficient is –5.
The third term is a constant. The degree of a nonzero polynomial
is the largest degree of any one term. In the example, the
polynomial has degree two.
Alternative forms
An expression that can be converted to polynomial
form through a sequence of applications of the commutative, associative, and distributive laws is
usually considered to be a polynomial. For instance
 (x+1)^3
is a polynomial because it can be worked out to
x^3+3x^2+3x+1. Similarly
 \frac
is considered a valid term in a polynomial, even
though it involves a division, because it is equivalent to
\tfracx^3 and \tfrac is just a constant. The coefficient of this
term is therefore \tfrac. For similar reasons, if complex
coefficients are allowed, one may have a single term like
(2+3i)x^3; even though it looks like it should be worked out to two
terms, the complex number 2+3i is in fact just a single coefficient
in this case that happens to require a "+" to be written
down.
Division by an expression containing a variable
is not generally allowed in polynomials. For example,
 \,
 ( 5 + y ) ^ x ,\,
Since subtraction can be treated as addition of
the additive opposite, and since exponentiation to a constant
positive whole number power can be treated as repeated
multiplication, polynomials can be constructed from constants and
variables with just the two operations addition and
multiplication.
Polynomial functions
A polynomial function is a function defined by
evaluating
a polynomial. A function ƒ of one argument is called a
polynomial function if it satisfies
 ƒ(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0
For example, the function f, taking real numbers
to real numbers, defined by
 f(x) = x^3  x
is a polynomial function of one argument.
Polynomial functions of multiple arguments can also be defined,
using polynomials in multiple variables, as in
 f(x,y)= 2x^3+4x^2y+xy^5+y^27.
Polynomial functions are an important class of
smooth
functions.
Polynomial equations
A polynomial equation is an equation in which a polynomial
is set equal to another polynomial.
 3x^2 + 4x 5 = 0 \,
is a polynomial equation. In case of a polynomial
equation the variable is considered an unknown, and one seeks to find
the possible values for which both members of the equation evaluate
to the same value (in general more than one solution may exist). A
polynomial equation is to be contrasted with a polynomial identity
like
(x + y)(x – y) = x2–y2,
where both members represent the same polynomial in different
forms, and as a consequence any evaluation of both members will
give a valid equality.
Elementary properties of polynomials
 A sum of polynomials is a polynomial
 A product of polynomials is a polynomial
 The derivative of a polynomial function is a polynomial function
 Any primitive or antiderivative of a polynomial function is a polynomial function
All polynomials have an expanded form, in which
the distributive
law has been used to remove all parentheses. All polynomials
with real or
complex
coefficients also have a factored form in which the polynomial is
written as a product of linear polynomials. For example, the
polynomial
 x^2  2x  3 \,
is the expanded form of the polynomial
 (x  3)(x + 1)\,,
which is written in factored form. Note that the
constants in the linear polynomials (like 3 and +1 in the above
example) may be complex
numbers in certain cases, even if all coefficients of the
expanded form are real numbers. This is because the field of real
numbers is not algebraically
closed; however, the
fundamental theorem of algebra states that the field of
complex
numbers is algebraically closed.
In school algebra, students learn to move easily
from one form to the other (see: factoring).
Every polynomial in one variable is equivalent to
a polynomial with the form
 a_n x^n + a_x^ + \cdots + a_2 x^2 + a_1 x + a_0.
This form is sometimes taken as the definition of
a polynomial in one variable.
Evaluation of a polynomial consists of assigning
a number to each variable and carrying out the indicated
multiplications and additions. Evaluation is sometimes performed
more efficiently using the Horner
scheme
 ((\ldots(a_n x + a_)x + ... + a_2)x + a_1)x + a_0\,.
In elementary algebra, methods are given for
solving all first degree and second degree polynomial equations in
one variable. In the case of polynomial equations, the variable is
often called an unknown. The number of solutions may not exceed the
degree, and will equal the degree when multiplicity of solutions
and complex
number solutions are counted. This fact is called the
fundamental theorem of algebra.
A system of polynomial equations is a set of
equations in which a given variable must take on the same value
everywhere it appears in any of the equations. Systems of equations
are usually grouped with a single open brace on the left. In
elementary
algebra, methods are given for solving a
system of linear equations in several unknowns. To get a unique
solution, the number of equations should equal the number of
unknowns. If there are more unknowns than equations, the system is
called
underdetermined. If there are more equations than unknowns, the
system is called overdetermined.
This important subject is studied extensively in the area of
mathematics known as linear
algebra. Overdetermined systems are common in practical
applications. For example, one U.S. mapping survey used computers
to solve 2.5 million equations in 400,000 unknowns.
More advanced examples of polynomials
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.
In abstract
algebra, one may define polynomials with coefficients in any
ring.
In knot theory
the Alexander
polynomial, the Jones
polynomial, and the HOMFLY
polynomial are important knot
invariants.
History
Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations are written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.Notation
The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La geometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.Solving polynomial equations
Every polynomial corresponds to a polynomial
function, where f(x) is set equal to the polynomial, and to a
polynomial equation, where the polynomial is set equal to zero. The
solutions to the equation are called the roots of the polynomial
and they are the zeroes of the function and the xintercepts of its
graph. If x = a is a root of a polynomial, then (x  a) is a factor
of that polynomial.
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 at least one
distinct root; this follows from the
fundamental theorem of algebra.
There is a difference between approximating roots
and finding exact roots. Formulas for the roots of polynomials up
to a
degree of 2 have been known since ancient times (see quadratic
equation) and up to a degree of 4 since the 16th century
(see Gerolamo
Cardano, Niccolo
Fontana Tartaglia). But formulas for degree 5 eluded
researchers. 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 relationships among
roots of polynomials.
Numerically solving a polynomial equation in one
unknown is easily done on computer by the DurandKerner
method or by some other rootfinding
algorithm. The reduction of equations in several unknowns to
equations each in one unknown is discussed in the article on the
Buchberger's
algorithm. The special case where all the polynomials are of
degree one is called a
system of linear equations, for which a range of different
solution methods exist, including the classical gaussian
elimination.
It has been shown by Richard
Birkeland and Karl Meyr that
the roots of any polynomial may be expressed in terms of
multivariate hypergeometric
functions. Ferdinand
von Lindemann and Hiroshi
Umemura showed that the roots may also be expressed in terms of
Siegel
modular functions, generalizations of the theta
functions that appear in the theory of elliptic
functions. These characterizations of the roots of arbitrary
polynomials are generalizations of the methods previously
discovered to solve the quintic
equation.
Graphs
A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial

 f(x) = 0
 is the xaxis.
 The graph of a degree 0 polynomial

 f(x) = a0 , where a0 ≠ 0,
 is a horizontal line with yintercept a0
 The graph of a degree 1 polynomial (or linear function)

 f(x) = a0 + a1x , where a1 ≠ 0,
 is an oblique line with yintercept a0 and slope a1.
 The graph of a degree 2 polynomial

 f(x) = a0 + a1x + a2x2, where a2 ≠ 0
 is a parabola.
 The graph of any polynomial with degree 2 or greater

 f(x) = a0 + a1x + a2x2 + . . . + anxn , where an ≠ 0 and n ≥ 2
 is a continuous nonlinear curve.
Polynomial graphs are analyzed in calculus using
intercepts, slopes, concavity, and end behavior.
The illustrations below show graphs of
polynomials.  x + 1   2  quadratic
 x^2 + 1   3  cubic 
x^3 + 1   4  quartic or biquadratic  x^4 + 1   5  quintic
 x^5 + 1   6  sextic or hexic  x^6 + 1   7  septic or
heptic  x^7 + 1   8  octic  x^8 + 1   9  nonic  x^9 + 1
  10  decic  x^ + 1  }
The names for degrees higher than 3 are less
common. The names for the degrees may be applied to the polynomial
or to its terms. For example, a constant may refer to a zero degree
polynomial or to a zero degree term.
The polynomial 0, which may be considered to have
no terms at all, is called the zero polynomial. Unlike other
constant polynomials, its degree is not zero. Rather the degree of
the zero polynomial is either left explicitly undefined, or defined
to be negative (either –1 or –∞)http://mathworld.wolfram.com/ZeroPolynomial.html.
The latter convention is important when defining Euclidean
division of polynomials.
The word monomial can be ambiguous, as it is also
often used to denote just a power of the variable, or in the
multivariate case product of such powers, without any coefficient.
Two or more terms which involve the same monomial in the latter
sense, in other words which differ only in the value of their
coefficients, are called similar terms; they can be combined into a
single term by adding their coefficients; if the resulting term has
coefficient zero, it may be removed altogether. The above
classification according to the number of terms assumes that
similar terms have been combined first.
Extensions of the concept of a polynomial
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.
 R. Birkeland. Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen. Mathematische Zeitschrift vol. 26, (1927) pp. 565578. Shows that the roots of any polynomial may be written in terms of multivariate hypergeometric functions. Paper is available here.
 F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen. Nachrichten von der Königl. Gesellschaft der Wissenschaften, vol. 7, 1884. Polynomial solutions in terms of theta functions. Paper available here.
 F. von Lindemann. Über die Auflösung der algebraischen Gleichungen durch transcendente Functionen II. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der GeorgAugustsUniversität zu Göttingen, 1892 edition. Paper available here.
 K. Mayr. Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen. Monatshefte für Mathematik und Physik vol. 45, (1937) pp. 280313.
 H. Umemura. Solution of algebraic equations in terms of theta constants. In D. Mumford, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston, 1984.
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