# 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[poly_{1},poly_{2},x] | find the result of dividing the polynomial in x by , dropping any remainder term |

PolynomialRemainder[poly_{1},poly_{2},x] | find the remainder from dividing the polynomial in x by |

PolynomialQuotientRemainder[poly_{1},poly_{2},x] |

| give the quotient and remainder in a list |

PolynomialMod[poly,m] | reduce the polynomial poly modulo m |

PolynomialGCD[poly_{1},poly_{2}] | find the greatest common divisor of two polynomials |

PolynomialLCM[poly_{1},poly_{2}] | find the least common multiple of two polynomials |

PolynomialExtendedGCD[poly_{1},poly_{2}] | find the extended greatest common divisor of two polynomials |

Resultant[poly_{1},poly_{2},x] | find the resultant of two polynomials |

Subresultants[poly_{1},poly_{2},x] | find the principal subresultant coefficients of two polynomials |

Discriminant[poly,x] | find the discriminant of the polynomial poly |

GroebnerBasis[{poly_{1},poly_{2},...},{x_{1},x_{2},...}] |

| find the Gröbner basis for the polynomials |

GroebnerBasis[{poly_{1},poly_{2},...},{x_{1},x_{2},...},{y_{1},y_{2},...}] |

| find the Gröbner basis eliminating the |

PolynomialReduce[poly,{poly_{1},poly_{2},...},{x_{1},x_{2},...}] |

| find a minimal representation of poly in terms of the |

Reduction of polynomials.

Given two polynomials

and

, one can always uniquely write

, where the degree of

is less than the degree of

.

PolynomialQuotient gives the quotient

, and

PolynomialRemainder gives the remainder

.

This gives the remainder from dividing

by

.

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Here is the quotient of

and

, with the remainder dropped.

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This gives back the original expression.

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Here the result depends on whether the polynomials are considered to be in

or

.

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PolynomialMod 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 appropriate multiples

of

m. The multiplier

q can itself be a polynomial, but its degree is always less than the degree of

poly.

PolynomialMod yields a final polynomial whose degree and leading coefficient are both as small as possible.

This reduces

modulo

. The result is simply the remainder from dividing the polynomials.

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The main difference between

PolynomialMod and

PolynomialRemainder 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

modulo both

and

.

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PolynomialGCD finds the highest degree polynomial that divides the

exactly. It gives the analog for polynomials of the integer function

GCD.

PolynomialGCD gives the greatest common divisor of the two polynomials.

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The returned polynomials

and

can be used to represent the GCD in terms of the original polynomials.

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The function

Resultant is used in a number of classical algebraic algorithms. The resultant of two polynomials

and

, 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 are zero.

Here is the resultant with respect to

of two polynomials in

and

. The original polynomials have a common root in

only for values of

at which the resultant vanishes.

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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.

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This polynomial has distinct roots, so its discriminant is nonzero.

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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.

The

is effectively redundant, and so does not appear in the Gröbner basis.

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The polynomial

has no roots, showing that the original polynomials have no common roots.

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The polynomials are effectively unwound here, and can now be seen to have exactly five common roots.

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PolynomialReduce yields a list

of polynomials with the property that

is minimal and

is exactly

poly.

This writes

in terms of

and

, leaving a remainder that depends only on

.

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Functions for factoring polynomials.

Factor,

FactorTerms, 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.

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FactorTerms pulls out only the factor of 2 that does not depend on

.

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FactorSquareFree factors out the 2 and the term

, but leaves the rest unfactored.

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Factor does full factoring, recovering the original form.

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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 for integers.

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.

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Factoring polynomials with complex coefficients.

Factor 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.

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When Gaussian integer coefficients are allowed, the polynomial factors.

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Irreducibility testing.

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.

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Over the Gaussian rationals, the polynomial is reducible.

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By default, algebraic numbers are treated as independent variables.

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Over the rationals extended by

Sqrt, the polynomial is reducible.

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Cyclotomic[n,x] | give the cyclotomic polynomial of order n in x |

Cyclotomic polynomials.

Cyclotomic polynomials arise as "elementary polynomials" in various algebraic algorithms. The cyclotomic polynomials are defined by

, where

runs over all positive integers less than

that are relatively prime to

.

This is the cyclotomic polynomial

.

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appears in the factors of

.

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Decompose[poly,x] | decompose poly, if possible, into a composition of a list of simpler polynomials |

Decomposing 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

of polynomials

. 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

, where

is the polynomial

.

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Here are two polynomial functions.

This gives the composition of the two functions.

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Decompose 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

and

, related by

and

give the same result on composition, so that

.

*Mathematica* 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.

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When

is

, the polynomial has value

.

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