The Factors function is a placeholder for representing the factorization of the multivariate polynomial a over U, a unique factorization domain. The construct Factors(a) produces a data structure of the form $\left[u\,\left[\left[f_1\,e_1\right]\,\mathrm{...}\,\left[f_n\,e_n\right]\right]\right]$ such that $a\=u{f}_1^{e_1}\cdots f_n^{e_n}$, where each f[i] is a primitive irreducible polynomial.

The difference between the Factors function and the Factor function is only the form of the result.  The Factor function, if defined, returns a Maple sum of products more suitable for interactive display and manipulation.

The call Factors(a) mod p computes the factorization of a over the integers modulo p, a prime integer. The polynomial a must have rational coefficients or coefficients over a finite field specified by RootOfs.

The call Factors(a, K) mod p computes the factorization over the finite field defined by K, an algebraic extension of the integers mod p where K is a RootOf.

The call modp1(Factors(a),p) computes the factorization of the polynomial a in the $\mathrm{modp1}$ representation modulo p a prime integer.

The call evala(Factors(a, K)) computes the factorization of the polynomial a over an algebraic number (or function) field defined by the extension K, which is specified as a RootOf or a set of RootOfs. The polynomial a must have algebraic number (or function) coefficients. The factors are monic for the ordering of the variables chosen by Maple.

Factors(2*x^2+6*x+6) mod 7;

$\left[2\,\left[\left[x+6\,1\right]\,\left[x+4\,1\right]\right]\right]$

Factors(x^5+1) mod 2;

$\left[1\,\left[\left[x+1\,1\right]\,\left[x^4+x^3+x^2+x+1\,1\right]\right]\right]$

alias(alpha=RootOf(x^2+x+1));

$\mathrm{\alpha }$

Factors(x^5+1,alpha) mod 2;

$\left[1\,\left[\left[\mathrm{\alpha }x+x^2+1\,1\right]\,\left[x+1\,1\right]\,\left[\mathrm{\alpha }x+x^2+x+1\,1\right]\right]\right]$

alias(sqrt2=RootOf(x^2-2)):

evala(Factors(2*x^2-1,sqrt2));

$\left[2\,\left[\left[x+\frac{\mathrm{sqrt2}}{2}\,1\right]\,\left[x-\frac{\mathrm{sqrt2}}{2}\,1\right]\right]\right]$

alias(sqrtx=RootOf(y^2-x,y)):

evala(Factors(x*y^2-1,sqrtx));

$\left[x\,\left[\left[y-\frac{\mathrm{sqrtx}}{x}\,1\right]\,\left[y+\frac{\mathrm{sqrtx}}{x}\,1\right]\right]\right]$

expand((x^3+y^5+2)*(x*y^2+3)) mod 7;

$x{y}^7+x^4{y}^2+3y^5+3x^3+2x{y}^2+6$

Factors((7)) mod 7;

$\left[1\,\left[\left[y^5+x^3+2\,1\right]\,\left[x{y}^2+3\,1\right]\right]\right]$

Factors(x^2+2*x*y+y^2+1+x+y,alpha) mod 5;

$\left[1\,\left[\left[y+x+4\mathrm{\alpha }\,1\right]\,\left[y+x+\mathrm{\alpha }+1\,1\right]\right]\right]$

Factors(x^2*y+x*y^2+2*alpha*x*y+alpha*x^2+4*alpha*x+y+alpha) mod 5;

$\left[1\,\left[\left[\mathrm{\alpha }x+xy+1\,1\right]\,\left[y+x+\mathrm{\alpha }\,1\right]\right]\right]$