The Rem and Quo functions are placeholders for representing the remainder and quotient respectively of a divided by b where a and b are polynomials in the variable x over a field.  They are used in conjunction with either  mod  or  $\mathrm{evala}$  as described below which define the coefficient domain.

Functionality:  Rem returns the remainder r and if the fourth argument q is present then the quotient is assigned to q. Quo returns the quotient q and if the fourth argument r is present then the remainder is assigned to r. The remainder r and quotient q satisfy:  $a=bq+r$.

The calls $\mathrm{Rem}\left(a\,b\,x\right)\mathbf{mod}p$ and $\mathrm{Quo}\left(a\,b\,x\right)\mathbf{mod}p$  compute the remainder and quotient respectively of a divided by b modulo p, a prime integer. The coefficients of a and b must be rational expressions over the rationals or over a finite field specified by RootOf expressions.  In particular, if the coefficients are integers then the computation is done over the field of integers modulo p.

The calls $\mathrm{evala}\left(\mathrm{Rem}\left(a\,b\,x\right)\right)$  and $\mathrm{evala}\left(\mathrm{Quo}\left(a\,b\,x\right)\right)$ compute the remainder and quotient respectively of a and b, where the coefficients of a and b are multivariate polynomials with coefficients in an algebraic number (or function) field.

$a≔x^4+5x^3+6\:$

$b≔x^2+2x+7\:$

$r≔\mathrm{Rem}\left(a\,b\,x\,'q'\right)\mathbf{mod}13$

$r≔5x+6$

$q$

$x^2+3x$

$\mathrm{Expand}\left(a-bq-r\right)\mathbf{mod}13$

$0$

$c≔x^2-x+3\:$

$d≔x-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\right)\:$

$\mathrm{evala}\left(\mathrm{Quo}\left(c\,d\,x\right)\right)$

$\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\right)+x-1$

$\mathrm{evala}\left(\mathrm{Rem}\left(c\,d\,x\right)\right)$

$-\mathrm{RootOf}\left({\mathrm{\_Z}}^2-3\right)+6$