The function prem  returns the pseudo-remainder r such that

where $\mathrm{degree}\left(r\&comma;x\right)<\mathrm{degree}\left(b\&comma;x\right)$ and m (the multiplier) is:

If the fourth argument is present it is assigned the value of the multiplier m defined above.  If the fifth argument is present, it is assigned the pseudo-quotient q defined above.

The function sprem has the same functionality as prem except that the multiplier m will be smaller, in general, equal to $\mathrm{lcoeff}\left(b\&comma;x\right)$ to the power of the number of division steps performed rather than the degree difference. If both $a$ and $b$ are multivariate polynomials with integer coefficients, then m is the (unique) smallest possible multiplier with positive leading coefficient that makes the pseudo-division fraction free.

When sprem can be used it is preferred over prem because it is more efficient.

$a≔x^4\&plus;1\&colon;$$b≔c{x}^2\&plus;1\&colon;$

$r≔\mathrm{prem}\left(a\&comma;b\&comma;x\&comma;\&apos;m\&apos;\&comma;\&apos;q\&apos;\right)\&colon;$

$r\&comma;m\&comma;q$

$c\left(c^2+1\right),c^3,c\left(c{x}^2-1\right)$

$r≔\mathrm{sprem}\left(a\&comma;b\&comma;x\&comma;\&apos;m\&apos;\&comma;\&apos;q\&apos;\right)\&colon;$

$r\&comma;m\&comma;q$

$c^2+1,c^2,c{x}^2-1$

$f≔4x^2\&plus;2x\&plus;1\&colon;$$g≔2x\&plus;1\&colon;$

$r≔\mathrm{prem}\left(f\&comma;g\&comma;x\&comma;\&apos;m\&apos;\&comma;\&apos;q\&apos;\right)\&colon;$

$r\&comma;m\&comma;q$

$4,4,8x$

$r≔\mathrm{sprem}\left(f\&comma;g\&comma;x\&comma;\&apos;m\&apos;\&comma;\&apos;q\&apos;\right)\&colon;$

$r\&comma;m\&comma;q$

$1,1,2x$