The built-in functions in this topic are for factoring numbers, polynomials, matrices, and Boolean expressions (for details, see Factoring NumbersFactoring_Numbers, Factoring PolynomialsFactoring_Polynomials, Factoring MatricesFactoring_Matrices, and Boolean FunctionsBoolean_Functions, respectively).  To collect expressions involving the elementary functions see Exponential FunctionsExponential_Functions, Logarithmic FunctionsLogarithmic_Functions, and Trigonometric FunctionsTrigonometric_Functions.


If n is a number, FACTOR(n, Number) factors n into its prime decomposition.  For example,

FACTOR(10!, Number)

simplifies to

 8  4  2   
2 ·3 ·5 ·7 


If u is a polynomial and amount is Trivial, Squarefree, Rational, Radical, or Complex, then FACTOR(u, amount) factors u with respect to all its variables.  FACTOR(u, amount, x, y, ...) factors u with respect to the variables x, y, ... .  If the amount argument is omitted, Rational is the default factoring amount for polynomials.  For example,

FACTOR(4·x^3 - 8·x^2 - 11·x - 3, Rational, x)

simplifies to

                 2 
(x - 3)·(2·x + 1)  


If u is a vector or matrix, FACTOR(u, Trivial) or just FACTOR(u) factors out the scalar content of the elements of u.


If u is a matrix, FACTOR(u, Turing) factors u to Turing form and FACTOR(u, Schmidt) factors u to Gram-Schmidt form.


If u is a Boolean expression, FACTOR(u) expands u to conjunctive normal form returning the result as a Boolean expression. For example,

FACTOR(p OR (q AND r))

simplifies to

(p ∨ q) ∧ (p ∨ r)


If n is a number, FACTORS(n) factors n into its prime decomposition (factorization) as a vector of pairs of primes and their powers.  For example, 

FACTORS(10!)

simplifies to [2, 8; 3, 4; 5, 2; 7, 1].


If u is a nonnumeric expression, FACTORS(u) returns the factors of u as a vector of base-degree pairs.  If u is a set or vector of expressions, FACTORS(u) distributes over the elements of u.  Note that sums are not necessarily factored.  For example,

FACTORS((x^2 - 1)·(x + 3)^2)

simplifies to [x^2 - 1, 1; x + 3, 2].


If u is a polynomial and amount is Trivial, Squarefree, Rational, Radical, or Complex, FACTORS(u, amount, x, y, ...) actively amount factors u by amount with respect to the variables x, y, ... and returns the result as a vector of base-degree pairs.  For example,

FACTORS((x^2 - 1)·(x + 3)^2, Rational)

simplifies to [x + 3, 2; x - 1, 1; x + 1, 1].


If u is a vector or matrix, FACTORS(u, Trivial) factors out the scalar content of the elements of u, and returns the result as a vector of base-degree pairs.


If u is a matrix, FACTORS(u, Turing) factors u to Turing form and FACTORS(u, Schmidt) factors u to Gram-Schmidt form, and returns the result as a vector of matrices.


If u is a Boolean expression, FACTORS(u) expands u to conjunctive normal form returning the result as a vector of disjuncts. For example,

FACTORS(p OR (q AND r))

simplifies to

[p ∨ q, p ∨ r]


Since FACTORS distributes over vectors, if v is a Boolean expression the composition FACTORS(TERMS(v)) simplifies to a vector of vectors of the literal atoms of the disjunctive normal form of v.  For example,

FACTORS(TERMS(NOT(NOT(a) OR b) OR NOT(c)))

simplifies to [[a, ¬ b], [¬ c]].


Other Built-in Functions and ConstantsBuilt_in_Functions_and_Constants 

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