Factor a polynomial into irreducible polynomials
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factor(f
, <Adjoin = adjoin
>, <MaxDegree = n
>) factor(f
,F  Domain = F  Full
)
factor(f)
computes a factorization f = u f_{1}^{e1} … f_{r}^{er} of
the polynomial f,
where u is
the content of f, f_{1},
…, f_{r} are
the distinct primitive irreducible factors of f,
and e_{1},
…, e_{r} are
positive integers.
factor
rewrites its argument as a product
of as many terms as possible. In a certain sense, it is the complementary
function of expand
,
which rewrites its argument as a sum of as many terms as possible.
If f
is a polynomial whose coefficient ring is not Expr
, then f
is
factored over its coefficient ring. See Example 10.
If f
is a polynomial with coefficient ring Expr
, then f
is
factored over the smallest ring containing the coefficients. Mathematically,
this implied coefficient ring always contains
the ring ℤ of
integers. See Example 4. If the
coefficient ring R
of f
is not Expr
, then
we say that the implied coefficient ring is R
.
Elements of the implied coefficient ring are considered to be constants
and are not factored any further. In particular, the content u is
an element of the implied coefficient ring.
With the option Adjoin
, the elements of adjoin
are
also adjoined to the coefficient ring.
If the second argument F
or, alternatively, Domain
= F
is given, then f is factored over the real numbers ℝ or
the complex numbers ℂ.
Factorization over ℝ or ℂ is
performed using numerical calculations and the results will contain
floatingpoint numbers. See Example 5.
If f
is an arithmetical expression but not a number, it is
considered as a rational expression. Nonrational subexpressions such
as sin(x)
, exp(1)
, x^(1/3)
etc.,
but not constant algebraic subexpressions such as I
and (sqrt(2)+1)^3
,
are replaced by auxiliary variables before factoring. Algebraic dependencies
of the subexpressions, such as the equation cos(x)^{2} =
1  sin(x)^{2},
are not necessarily taken into account. See Example 7.
The resulting expression is then written as a quotient of two polynomial expressions in
the original and the auxiliary indeterminates. The numerator and the
denominator are converted into polynomials with coefficient ring Expr
via poly
, and the implied
coefficient ring is the smallest ring containing the coefficients
of the numerator polynomial and the denominator polynomial. Usually,
this is the ring of integers. Then both polynomials are factored over
the implied coefficient ring, and the multiplicities e_{i} corresponding
to factors of the denominator are negative integers; see Example 3. After the factorization,
the auxiliary variables are replaced by the original subexpressions.
See Example 6.
If f
is an integer,
then it is decomposed into a product of primes, and the result is
the same as for ifactor
.
If f
is a rational number,
then both the numerator and the denominator are decomposed into a
product of primes. In this case, the multiplicities e_{i} corresponding
to factors of the denominator are negative integers. See Example 2.
If f
is a floating
point number or a complex number,
then factor
returns a factorization with the single
factor f
.
The result of factor
is an object of the
domain type Factored
.
Let g:=factor(f)
be such an object.
It is represented internally by the list[u,
f1, e1, ..., fr, er]
of odd length 2 r +
1. Here, f1
through fr
are
of the same type as the input (either polynomials or expressions); e1
through er
are
integers; and u
is an arithmetical expression.
One may extract the content u and
the terms f_{i}^{ei} by
the ordinary index operator [ ]
, i.e., g[1] = f1^e1, g[2]
= e1^e2, ...
if u =
1 and g[1] = u, g[2] = f1^e1, g[3] =
e1^e2, ...
, respectively, if u ≠
1.
The call Factored::factors(g)
yields the
list [f1, f2, ...]
of factors, the call Factored::exponents(g)
returns
the list [e1, e2, ...]
of exponents.
The call coerce
(g,DOM_LIST)
returns
the internal representation of a factored object, i.e., the list [u,
f1, e1, f2, e2, ...]
.
Note that the result of factor
is printed
as an expression, and it is implicitly converted into an expression
whenever it is processed further by other MuPAD^{®} functions. As
an example, the result of q:=factor(x^2+2*x+1)
is
printed as (x+1)^2
, which is an expression of type "_power"
.
See Example 1 for
illustrations, and the help page of Factored
for details.
If f
is not a number, then each of the polynomials p_{1},
…, p_{r} is
primitive, i.e., the greatest common divisor of its coefficients (see content
and gcd
) over the implied
coefficient ring (see above for a definition) is one.
Currently, factoring polynomials is possible over the following
implied coefficient rings: integers, real numbers, complex numbers
and rational numbers, finite fields—represented by IntMod(n)
or Dom::IntegerMod(n)
for
a prime number n
, or by a Dom::GaloisField
—, and rings obtained
from these basic rings by taking polynomial rings (see Dom::DistributedPolynomial
, Dom::MultivariatePolynomial
, Dom::Polynomial
, and Dom::UnivariatePolynomial
),
fields of fractions (see Dom::Fraction
),
and algebraic extensions (see Dom::AlgebraicExtension
).
If the input f
is an arithmetical expression that
is not a number,
all occurring floatingpoint numbers are replaced by continued fraction
approximations. The result is sensitive to the environment variable DIGITS
,
see numeric::rationalize
for
details.
To factor the polynomial x^{3} + x, enter:
g := factor(x^3+x)
Usually, expressions are factored over the ring of integers,
and factors with nonintegral coefficients, such as x  I
in
the example above, are not considered.
One can access the internal representation of this factorization with the ordinary index operator:
g[1], g[2]
The internal representation of g
, as described
above, is given by the following command:
coerce(g, DOM_LIST)
The result of the factorization is an object of domain type Factored
:
domtype(g)
Some of the functionality of this domain is described in what follows.
One may extract the factors and exponents of the factorization also in the following way:
Factored::factors(g), Factored::exponents(g)
One can ask for the type of factorization:
Factored::getType(g)
This output means that all f_{i} are
irreducible. Other possible types are "squarefree"
(see polylib::sqrfree
) or "unknown"
.
One may multiply factored objects, which preserves the factored form:
g2 := factor(x^2 + 2*x + 1)
g * g2
It is important to note that one can apply (almost) any function
working with arithmetical expressions to an object of type Factored
.
However, the result is then usually not of domain type Factored
:
expand(g); domtype(%)
For a detailed description of these objects, please refer to
the help page of the domain Factored
.
factor
splits an integer into a product of
prime factors:
factor(8)
For rational numbers, both the numerator and the denominator are factored:
factor(10/33)
Note that, in contrast, constant polynomials are not factored:
factor(poly(8, [x]))
Factors of the denominator are indicated by negative multiplicities:
factor((z^2  1)/z^2)
Factored::factors(%), Factored::exponents(%)
If some coefficients are irrational but algebraic, the factorization
takes place over the smallest field extension of the rationals that
contains all of them. Hence, x^2+1
is considered
irreducible while its I
fold is considered reducible:
factor(x^2 + 1), factor(I*x^2 + I)
MuPAD does not automatically factor over the field of algebraic numbers; only the coefficients of the input are adjoined to the rationals:
factor(sqrt(2)*x^4  sqrt(2)*x^2  sqrt(2)*2)
factor(I*x^4  I*x^2  I*2)
factor(sqrt(2)*I*x^4  sqrt(2)*I*x^2  sqrt(2)*I*2)
With the option Adjoin, additional elements can be adjoined to the implied coefficient ring:
factor(x^2 + 1, Adjoin = [I])
factor( x^22, Adjoin = {sqrt(2)} )
With the option Full
, a complete factorization
into linear factors can be computed.
factor( x^22, Full)
If the argument R_ or C_ is given, factorization is done over the real or complex numbers using numeric calculations:
factor( x^22, R_ )
factor(x^2 + 1, C_)
Transcendental objects are treated as indeterminates:
delete x: factor(7*(cos(x)^2  1)*sin(1)^3)
Factored::factors(%), Factored::exponents(%)
factor
regards transcendental subexpressions
as algebraically independent of each other. Sometimes, the dependence
is recognized:
factor(x + 2*sqrt(x) + 1)
In many cases, however, the algebraic dependence is not recognized:
factor(x^2 + (2^y*3^y + 6^y)* x + (6^y)^2)
factor
replaces floatingpoint numbers by
continued fraction approximations, factors the resulting polynomial,
and finally applies float
to
the coefficients of the factors:
factor(x^2 + 2.0*x  8.0)
factor
with the option Full
can
use RootOf
to
symbolically represent the roots of a polynomial:
factor(x^5 + x^2 + 1, Full)
Polynomials with a coefficient ring other than Expr
are factored over
their coefficient ring. We factor the following polynomial modulo 17:
R := Dom::IntegerMod(17): f:= poly(x^3 + x + 1, R): factor(f)
For every p
, the expression IntMod(p)
may
be used instead of Dom::IntegerMod
(p)
:
R := IntMod(17): f:= poly(x^3 + x + 1, R): factor(f)
More complex domains are allowed as coefficient rings, provided they can be obtained from the rational numbers or from a finite field by iterated construction of algebraic extensions, polynomial rings, and fields of fractions. In the following example, we factor the univariate polynomial u^{2}  x^{3} in u over the coefficient field :
Q := Dom::Rational: Qx := Dom::Fraction(Dom::DistributedPolynomial([x], Q)): F := Dom::AlgebraicExtension(Qx, poly(z^2  x, [z])): f := poly(u^2  x^3, [u], F)
factor(f)

A polynomial or an arithmetical expression 

R_ or C_ 

Option, specified as Only algebraic numbers of a maximum degree 

Option, specified as In addition to the coefficients of 

Option, specified as Compute a numerical factorization over or , respectively. 

Compute the full factorization of 
Object of the domain type Factored
.
f
The factoring algorithms are collected in a separate library
domain faclib
; it should not be necessary to call
these routines directly.
The implemented algorithms include CantorZassenhaus (over finite fields) and Hensel lifting (over the rational numbers and in the multivariate case).
collect
 content
 denom
 div
 divide
 expand
 Factored
 gcd
 icontent
 ifactor
 igcd
 ilcm
 indets
 irreducible
 isprime
 lcm
 normal
 numer
 partfrac
 polylib::decompose
 polylib::divisors
 polylib::primpart
 polylib::sqrfree
 rationalize
 simplify