An expression of the form
where
are variables and
(the
coefficients
of the polynomial) and
(the
exponents of the powers,
which are non-negative integers) are constants. The individual expressions
are called the
terms of the polynomial.
The order of the terms, and also the order of the factors in each
term, can be changed arbitrarily; in precisely the same way it is
possible to introduce or omit terms with zero coefficients and, in each
individual term, zero powers. When the polynomial has one, two or
three terms it is called a monomial, binomial or trinomial.
With regard to the coefficients of a polynomial one assumes that they belong to a
field,
for example, the field of rational, real or complex numbers.
Two terms of a polynomial are called
similar
if the powers of the same variables in them are equal. Terms similar to each other,
can be replaced by one term
(reduction of similar terms).
Two polynomials are called
equal
if, after reduction, all terms with non-zero coefficients are
pairwise identical (but, possibly, written in a different order), and also if
all the coefficients of both of these polynomials turn out to
be zero. In the latter case the polynomial is called
identically zero
and is denoted by the symbol 0.
The sum of the powers of any term of a polynomial is called the
degree
of that term. If the polynomial is not identically zero,
then among the terms with non-zero coefficients (it is assumed that similar
terms have been reduced) there is at least one
of highest degree: this highest degree is called the
degree of the polynomial.
The zero polynomial does not have a degree. A
polynomial of degree zero reduces to a single term
(a constant, not equal to zero).
A polynomial in the variables
is called a
symmetric polynomial
if it is not changed by any permutation of the
variables. A polynomial of which all terms have the same degree is called a
homogeneous polynomial
or a
form;
forms of the first, second or third degree are called
linear, quadratic or cubic, and, according to the number of variables
(two or three), they are called dyadic (binary) or triadic (ternary) (for example,
is a ternary quadratic form).
The
degree
of a polynomial
with respect to one of its variables
,
,
is the highest power with which
occurs in a term of this polynomial (this degree may be
zero). Of two terms of a polynomial the higher one (relative to
a given numbering of the variables) is that for which the power of
is higher, and if these powers are equal, that for which the power of
is higher, etc. If all terms of a polynomial are ordered so that each
term is lower than the preceding, then the terms are said to be
lexicographically ordered.
The term which then stands in the first place is called the
highest term
(or
leading term).
A polynomial of one variable with lexicographically ordered terms has the form
where
are the coefficients.
The
roots of a polynomial
in one variable over a field
are the solutions of the
algebraic equation
The roots of a polynomial are related to its coefficients by Viète's formula (see
Viète theorem).
The set of all possible polynomials in
variables with coefficients from a given field forms a
ring
with respect to the naturally defined operations of addition and multiplication. The
ring of polynomials in an infinite set of variables can also be considered. A
ring of polynomials
is an associative-commutative ring without zero divisors (that is,
a product of non-zero polynomials cannot be 0).
If for two given polynomials
and
there exists a polynomial
such that
,
then one says that
is divisible by
;
is called the divisor and
the quotient. If
is not divisible by
,
but both polynomials contain the same variable, for example
,
and the degree of
relative to
is
and the degree of
relative to
is
,
,
then there are polynomials
,
and
such that
,
where
does not contain
at all and
occurs in
with degree less than
.
When
is the only variable, then
can be taken to be 1; in this case the operation of finding
and
from
and
is called
division with remainder;
division with remainder can be carried out using the
Horner scheme.
By repeated application of this operation it is
possible to find the greatest common divisor of
and
,
that is, the divisor of
and
which is divisible by any common divisor of these polynomials (see
Euclidean algorithm).
Two polynomials with greatest common divisor equal to 1 are called
coprime.
A polynomial which can be represented as a product of polynomials
of smaller degree with coefficients from a given field is called
reducible
(over that field); otherwise it is called
irreducible.
The irreducible polynomials play a role in the ring of
polynomials similar to that played by the prime numbers in the
ring of integers. For example, the following theorem holds: If a product
is divisible by an irreducible polynomial
and
is not divisible by
,
then
must be divisible by
.
Each polynomial of degree greater than zero splits over a given field into
a product of irreducible factors in a unique way (up
to factors of degree zero). For example, the polynomial
is irreducible over the field of rational numbers, splits into two factors
over the field of real numbers and into four factors over
the field of complex numbers. In general, each polynomial of one variable
with real coefficients splits over the field of real numbers into factors
of the first and second degree, and over the field
of complex numbers into factors of the first degree (cf.
Algebra, fundamental theorem of).
For two or more variables this is no longer true. Over any field
,
for
there are polynomials in
variables that are irreducible over any extension of
.
Such polynomials are called
absolutely irreducible.
For example, the polynomial
is irreducible over any number field.
If the variables
are given numerical values (for example, real or complex), then the polynomial assumes
a certain numerical value. Thus, a polynomial can be considered as
a function of the corresponding variables. This function is continuous and
differentiable for any values of the variables; it can be characterized as an
entire rational function,
that is, a function obtained from variables and constants (the coefficients)
by performing in a specific order the operations of addition,
subtraction and multiplication. Entire rational functions belong to the broader class of
rational functions,
where division is added to the list of operations:
Any rational function can be represented as a
quotient of two polynomials. Finally, rational functions are
contained in the class of algebraic functions (cf.
Algebraic function).
One of the most important properties of polynomials is that
any continuous function on a compact subset of the
complex plane can be approximated by a polynomial within arbitrarily small error (see
Weierstrass theorem).
Special systems of polynomials,
orthogonal polynomials,
are used in approximation theory as a means of representing functions by series.