# 9.22 Factored¶

Factored creates a domain whose objects are kept in factored form as long as possible. Thus certain operations like * (multiplication) and gcdgcdFactored are relatively easy to do. Others, such as addition, require somewhat more work, and the result may not be completely factored unless the argument domain R provides a factorfactorFactored operation. Each object consists of a unit and a list of factors, where each factor consists of a member of R (the base), an exponent, and a flag indicating what is known about the base. A flag may be one of nil, sqfr, irred or prime, which mean that nothing is known about the base, it is square-free, it is irreducible, or it is prime, respectively. The current restriction to factored objects of integral domains allows simplification to be performed without worrying about multiplication order.

## 9.22.1 Decomposing Factored Objects¶

In this section we will work with a factored integer.

g := factor(4312)


 237211

Type: Factored Integer

Let’s begin by decomposing g into pieces. The only possible units for integers are 1 and -1.

unit(g)


 1

Type: PositiveInteger

There are three factors.

numberOfFactors(g)


 3

Type: PositiveInteger

We can make a list of the bases, ...

[nthFactor(g,i) for i in 1..numberOfFactors(g)]


 [2,7,11]

Type: List Integer

and the exponents, ...

[nthExponent(g,i) for i in 1..numberOfFactors(g)]


 [3,2,1]

Type: List Integer

and the flags. You can see that all the bases (factors) are prime.

[nthFlag(g,i) for i in 1..numberOfFactors(g)]


 [“prime”,”prime”,”prime”]

Type: List Union(“nil”,”sqfr”,”irred”,”prime”)

A useful operation for pulling apart a factored object into a list of records of the components is factorListfactorListFactored.

factorList(g)


 [[flg=”prime”,fctr=2,xpnt=3],[flg=”prime”,fctr=7,xpnt=2],[flg=”prime”,fctr=11,xpnt=1]]

Type: List Record(flg: Union(“nil”,”sqfr”,”irred”,”prime”), fctr: Integer,xpnt: Integer)

If you don’t care about the flags, use factorsfactorsFactored.

factors(g)


 [[factor=2,exponent=3],[factor=7,exponent=2],[factor=11,exponent=1]]

Type: List Record(factor: Integer,exponent: Integer)

Neither of these operations returns the unit.

first(%).factor


 2

Type: PositiveInteger

## 9.22.2 Expanding Factored Objects¶

Recall that we are working with this factored integer.

g := factor(4312)


 237211

Type: Factored Integer

To multiply out the factors with their multiplicities, use expandexpandFactored.

expand(g)


 4312

Type: PositiveInteger

If you would like, say, the distinct factors multiplied together but with multiplicity one, you could do it this way.

reduce(*,[t.factor for t in factors(g)])


 154

Type: PositiveInteger

## 9.22.3 Arithmetic with Factored Objects¶

We’re still working with this factored integer.

g := factor(4312)


 237211

Type: Factored Integer

We’ll also define this factored integer.

f := factor(246960)


 2432573

Type: Factored Integer

Operations involving multiplication and division are particularly easy with factored objects.

f * g


 273257511

Type: Factored Integer

f^500


 2200031000550071500

Type: Factored Integer

gcd(f,g)


 2372

Type: Factored Integer

lcm(f,g)


 243257311

Type: Factored Integer

If we use addition and subtraction things can slow down because we may need to compute greatest common divisors.

f + g


 2372641

Type: Factored Integer

f - g


 2372619

Type: Factored Integer

Test for equality with 0 and 1 by using zero?zero?Factored and one?one?Factored, respectively.

zero?(factor(0))


 true

Type: Boolean

zero?(g)


 false

Type: Boolean

one?(factor(1))


 true

Type: Boolean

one?(f)


 false

Type: Boolean

Another way to get the zero and one factored objects is to use package calling (see ugTypesPkgCallPage in Section ugTypesPkgCallNumber ).

0$Factored(Integer)    0 Type: Factored Integer 1$Factored(Integer)


 1

Type: Factored Integer

## 9.22.4 Creating New Factored Objects¶

The mapmapFactored operation is used to iterate across the unit and bases of a factored object. See FactoredFunctionsTwoXmpPage for a discussion of mapmapFactored.

The following four operations take a base and an exponent and create a factored object. They differ in handling the flag component.

nilFactor(24,2)


 242

Type: Factored Integer

This factor has no associated information.

nthFlag(%,1)


 “nil”

Type: Union(“nil”,...)

This factor is asserted to be square-free.

sqfrFactor(30,2)


 302

Type: Factored Integer

This factor is asserted to be irreducible.

irreducibleFactor(13,10)


 1310

Type: Factored Integer

This factor is asserted to be prime.

primeFactor(11,5)


 115

Type: Factored Integer

A partial inverse to factorListfactorListFactored is makeFRmakeFRFactored.

h := factor(-720)


 -24325

Type: Factored Integer

The first argument is the unit and the second is a list of records as returned by factorListfactorListFactored.

h - makeFR(unit(h),factorList(h))


 0

Type: Factored Integer

## 9.22.5 Factored Objects with Variables¶

Some of the operations available for polynomials are also available for factored polynomials.

p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x +


63


 (4x2-12x+9)y2+(4x2-12x+9)y+28x2-84x+63

Type: Polynomial Integer

fp := factor(p)


 (2x-3)2(y2+y+7)

Type: Factored Polynomial Integer

You can differentiate with respect to a variable.

D(p,x)


 (8x-12)y2+(8x-12)y+56x-84

Type: Polynomial Integer

D(fp,x)


 4(2x-3)(y2+y+7)

Type: Factored Polynomial Integer

numberOfFactors(%)


 3

Type: PositiveInteger