Important: The polytools package has been deprecated. Use the superseding command PolynomialTools[Splits] instead.

The splits function computes the complete factorization of a univariate polynomial.

The univariate polynomial a in x, with rational or algebraic number coefficients or polynomial coefficients, is factored into linear factors.

The factorization is returned in the form $\left[u\,\left[\left[f_1\,m_1\right]\,..\,\left[f_n\,m_n\right]\right]\right]$ where $a=u{f_1}^{m_1}\mathrm{...}{f_n}^{m_n}$ where each $f_k$ (the factor) is a unit normal irreducible polynomial and each $m_k$ (its multiplicity) is a positive integer.

In the case of three arguments, the third argument b is assigned the extensions of Q on output, where Q is the set of rational numbers.

The command with(polytools,splits) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{polytools}\right)\:$

$\mathrm{splits}\left(x^2+x+1\,x\right)$

$\left[1\,\left[\left[x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+1\right)\,1\right]\,\left[x+1+\mathrm{RootOf}\left({\mathrm{\_Z}}^2+\mathrm{\_Z}+1\right)\,1\right]\right]\right]$

$\mathrm{splits}\left(x^2+yx+1+y^2\,x\,'b'\right)$

$\left[1\,\left[\left[x+y+\mathrm{RootOf}\left({\mathrm{\_Z}}^2+y\mathrm{\_Z}+y^2+1\right)\,1\right]\,\left[x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2+y\mathrm{\_Z}+y^2+1\right)\,1\right]\right]\right]$

$b$

$\left\{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+y\mathrm{\_Z}+y^2+1\right)\right\}$