Main Index
Algebraic structures
Polynomial rings
Univariate polynomials
Subject Index
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Main Index
Algebraic structures
Polynomial rings
Univariate polynomials
Subject Index
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A (univariate) polynomial 
in 
is an expression of the form
![n n - 1 FormBox[Cell[TextData[Cell[Box ... TraditionalForm] n n - 1 1 0](HTMLFiles/PolynomialsBasic_3.gif){kind=small}
where 
are constants, called coefficients and 
the so-called indeterminate. If the coefficients are admitted to be elements of a set 
, then 
denotes the set of the all polynomials with coefficient from the set 
. The number 
is called the degree of 
(or simply 
) provided 
, and denoted by 
. The coefficient 
is called leading coefficient of 
and will be denoted by 
. The coefficient 
is called the constant or absolute term of the polynomial. The expressions 
, 
, are called monomials or terms of 
.
The polynomials of the zero degree are called constant, of degree two quadratic, of degree three cubic, of degree four biquadratic, etc. Polynomials of the form 
are sometimes called pure polynomials. Polynomials with the leading coefficient 
are called monic.
The polynomial whit all the coefficient vanishing is called the zero polynomial and will be denoted by 
. It is convenient to assign the zero polynomial
Example. The expressions 
are actually polynomials of degree 
in 
with integer coefficients. They are called Chebyshev’s polynomials.
If 
and 
are arbitrary polynomials in 
, then 
and 
.
Manipulation with polynomials is governed by the following rules. If
 |  |
 |  |
are two polynomials in 
with the assumption 
then we have:
Equality: 
if and only if 
for 
, and 
.
Addition: 
, where 
for 
and 
for 
.
Multiplication: 
where 
for 
.
Cite this web-page as:
Štefan Porubský: Polynomials Basic.