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.

$\mathrm{Factors}\left(2x^2+6x+6\right)\mathbf{mod}7$

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

$\mathrm{Factors}\left(x^5+1\right)\mathbf{mod}2$

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

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(x^2+x+1\right)\right)$

$\mathrm{\alpha }$

$\mathrm{Factors}\left(x^5+1\,\mathrm{\alpha }\right)\mathbf{mod}2$

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

$\mathrm{alias}\left(\mathrm{sqrt2}=\mathrm{RootOf}\left(x^2-2\right)\right)\:$

$\mathrm{evala}\left(\mathrm{Factors}\left(2x^2-1\,\mathrm{sqrt2}\right)\right)$

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

$\mathrm{alias}\left(\mathrm{sqrtx}=\mathrm{RootOf}\left(y^2-x\,y\right)\right)\:$

$\mathrm{evala}\left(\mathrm{Factors}\left(x{y}^2-1\,\mathrm{sqrtx}\right)\right)$

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

$\mathrm{expand}\left(\left(x^3+y^5+2\right)\left(x{y}^2+3\right)\right)\mathbf{mod}7$

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

$\mathrm{Factors}\left(\right)\mathbf{mod}7$

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

$\mathrm{Factors}\left(x^2+2xy+y^2+1+x+y\,\mathrm{\alpha }\right)\mathbf{mod}5$

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

$\mathrm{Factors}\left(x^{2}y+x{y}^2+2\mathrm{\alpha }xy+\mathrm{\alpha }{x}^2+4\mathrm{\alpha }x+y+\mathrm{\alpha }\right)\mathbf{mod}5$

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