Algebraic Operations on Polynomials
For many kinds of practical calculations, the only operations you will need to perform on polynomials are essentially structural ones.
If you do more advanced algebra with polynomials, however, you will have to use the algebraic operations discussed in this tutorial.
You should realize that most of the operations discussed in this tutorial work only on ordinary polynomials, with integer exponents and rational-number coefficients for each term.
|PolynomialQuotient[poly1,poly2,x]||find the result of dividing the polynomial in x by , dropping any remainder term|
|PolynomialRemainder[poly1,poly2,x]||find the remainder from dividing the polynomial in x by |
|give the quotient and remainder in a list|
|PolynomialMod[poly,m]||reduce the polynomial poly modulo m|
|PolynomialGCD[poly1,poly2]||find the greatest common divisor of two polynomials|
|PolynomialLCM[poly1,poly2]||find the least common multiple of two polynomials|
|PolynomialExtendedGCD[poly1,poly2]||find the extended greatest common divisor of two polynomials|
|Resultant[poly1,poly2,x]||find the resultant of two polynomials|
|Subresultants[poly1,poly2,x]||find the principal subresultant coefficients of two polynomials|
|Discriminant[poly,x]||find the discriminant of the polynomial poly|
|find the Gröbner basis for the polynomials |
|find the Gröbner basis eliminating the |
|find a minimal representation of poly in terms of the |
Reduction of polynomials.
Given two polynomials
, one can always uniquely write
, where the degree of
is less than the degree of
gives the quotient
, and PolynomialRemainder
gives the remainder
This gives the remainder from dividing
Here is the quotient of
, with the remainder dropped.
This gives back the original expression.
Here the result depends on whether the polynomials are considered to be in
is essentially the analog for polynomials of the function Mod
for integers. When the modulus m
is an integer, PolynomialMod
simply reduces each coefficient in poly
modulo the integer m
. If m
is a polynomial, then PolynomialMod
effectively tries to get a polynomial with as low a degree as possible by subtracting from poly
. The multiplier q
can itself be a polynomial, but its degree is always less than the degree of poly
yields a final polynomial whose degree and leading coefficient are both as small as possible.
. The result is simply the remainder from dividing the polynomials.
The main difference between PolynomialMod
is that while the former works simply by multiplying and subtracting polynomials, the latter uses division in getting its results. In addition, PolynomialMod
allows reduction by several moduli at the same time. A typical case is reduction modulo both a polynomial and an integer.
This reduces the polynomial
finds the highest degree polynomial that divides the
exactly. It gives the analog for polynomials of the integer function GCD
gives the greatest common divisor of the two polynomials.
The returned polynomials
can be used to represent the GCD in terms of the original polynomials.
The function Resultant
is used in a number of classical algebraic algorithms. The resultant of two polynomials
, both with leading coefficient one, is given by the product of all the differences
between the roots of the polynomials. It turns out that for any pair of polynomials, the resultant is always a polynomial in their coefficients. By looking at when the resultant is zero, you can tell for what values of their parameters two polynomials have a common root. Two polynomials with leading coefficient one have
common roots if exactly the first
elements in the list Subresultants
Here is the resultant with respect to
of two polynomials in
. The original polynomials have a common root in
only for values of
at which the resultant vanishes.
The function Discriminant
is the product of the squares of the differences of its roots. It can be used to determine whether the polynomial has any repeated roots. The discriminant is equal to the resultant of the polynomial and its derivative, up to a factor independent of the variable.
This polynomial has a repeated root, so its discriminant vanishes.
This polynomial has distinct roots, so its discriminant is nonzero.
Gröbner bases appear in many modern algebraic algorithms and applications. The function GroebnerBasis
takes a set of polynomials, and reduces this set to a canonical form from which many properties can conveniently be deduced. An important feature is that the set of polynomials obtained from GroebnerBasis
always has exactly the same collection of common roots as the original set.
is effectively redundant, and so does not appear in the Gröbner basis.
has no roots, showing that the original polynomials have no common roots.
The polynomials are effectively unwound here, and can now be seen to have exactly five common roots.
yields a list
of polynomials with the property that
is minimal and
is exactly poly
in terms of
, leaving a remainder that depends only on
Functions for factoring polynomials.
, and FactorSquareFree
perform various degrees of factoring on polynomials. Factor
does full factoring over the integers. FactorTerms
extracts the "content" of the polynomial. FactorSquareFree
pulls out any multiple factors that appear.
Here is a polynomial, in expanded form.
pulls out only the factor of 2 that does not depend on
factors out the 2 and the term
, but leaves the rest unfactored.
does full factoring, recovering the original form.
Particularly when you write programs that work with polynomials, you will often find it convenient to pick out pieces of polynomials in a standard form. The function FactorList
gives a list of all the factors of a polynomial, together with their exponents. The first element of the list is always the overall numerical factor for the polynomial.
The form that FactorList
returns is the analog for polynomials of the form produced by FactorInteger
Here is a list of the factors of the polynomial in the previous set of examples. Each element of the list gives the factor, together with its exponent.
Factoring polynomials with complex coefficients.
and related functions usually handle only polynomials with ordinary integer or rational-number coefficients. If you set the option GaussianIntegers->True
, however, then Factor
will allow polynomials with coefficients that are complex numbers with rational real and imaginary parts. This often allows more extensive factorization to be performed.
This polynomial is irreducible when only ordinary integers are allowed.
When Gaussian integer coefficients are allowed, the polynomial factors.
A polynomial is irreducible over a field
if it cannot be represented as a product of two nonconstant polynomials with coefficients in
This polynomial is irreducible over the rationals.
Over the Gaussian rationals, the polynomial is reducible.
By default, algebraic numbers are treated as independent variables.
Over the rationals extended by Sqrt
, the polynomial is reducible.
|Cyclotomic[n,x]||give the cyclotomic polynomial of order n in x|
Cyclotomic polynomials arise as "elementary polynomials" in various algebraic algorithms. The cyclotomic polynomials are defined by
runs over all positive integers less than
that are relatively prime to
This is the cyclotomic polynomial
appears in the factors of
|Decompose[poly,x]||decompose poly, if possible, into a composition of a list of simpler polynomials|
Factorization is one important way of breaking down polynomials into simpler parts. Another, quite different, way is decomposition
. When you factor a polynomial
, you write it as a product
. Decomposing a polynomial
consists of writing it as a composition
of polynomials of the form
Here is a simple example of Decompose
. The original polynomial
can be written as the polynomial
is the polynomial
Here are two polynomial functions.
This gives the composition of the two functions.
is set up to give a list of polynomials in
, which, if composed, reproduce the original polynomial. The original polynomial can contain variables other than
, but the sequence of polynomials that Decompose
produces are all intended to be considered as functions of
Unlike factoring, the decomposition of polynomials is not completely unique. For example, the two sets of polynomials
, related by
give the same result on composition, so that
follows the convention of absorbing any constant terms into the first polynomial in the list produced by Decompose
Generating interpolating polynomials.
This yields a quadratic polynomial which goes through the specified three points.
, the polynomial has value