The Quotient['right'](Poly1, Poly2, A) or Quotient(Poly1, Poly2, A) calling sequence returns the right quotient Q of Poly1 and Poly2 such that:

where the degree of the right remainder R is less than that of Poly2.

If the fourth argument 'R' is specified, it is assigned the right remainder defined above.

The Quotient['left'](Poly1, Poly2, A) calling sequence returns the right quotient Q of Poly1 and Poly2 such that:

where the degree of the left remainder R is less than that of Poly2.

If the fourth argument 'R' is specified, it is assigned the left remainder defined above.

The Remainder['right'](Poly1, Poly2, A) or Remainder(Poly1, Poly2, A) calling sequence returns the right remainder R of Poly1 and Poly2 such that:

where the degree of R is less than that of Poly2 and Q is the right quotient.

If the fourth argument 'Q' is specified, it is assigned the right quotient defined above.

The Remainder['left'](Poly1, Poly2, A) calling sequence returns the left remainder R of Poly1 and Poly2 such that:

where the degree of R is less than that of Poly2 and Q is the left quotient.

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

$A≔\mathrm{SetOreRing}\left(x\,'\mathrm{differential}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(x\,\mathrm{differential}\right)$

$\mathrm{Poly1}≔\mathrm{OrePoly}\left(\frac{1}{x}\,0\,\frac{x}{x+1}\,\frac{x^2}{x+2}\right)$

$\mathrm{Poly1}≔\mathrm{OrePoly}\left(\frac{1}{x}\,0\,\frac{x}{x+1}\,\frac{x^2}{x+2}\right)$

$\mathrm{Poly2}≔\mathrm{OrePoly}\left(x\,\frac{x+1}{x}\,x\right)$

$\mathrm{Poly2}≔\mathrm{OrePoly}\left(x\,\frac{x+1}{x}\,x\right)$

$\mathrm{R1}≔\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\right)$

$\mathrm{R1}≔\mathrm{OrePoly}\left(\frac{x^2+4x+2}{x\left(x+1\right)\left(x+2\right)}\,-\frac{x^4-x^2-2x-1}{\left(x+2\right)x^2}\right)$

$\mathrm{R2}≔\mathrm{Remainder}\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\right)$

$\mathrm{R2}≔\mathrm{OrePoly}\left(\frac{x^2+4x+2}{x\left(x+1\right)\left(x+2\right)}\,-\frac{x^4-x^2-2x-1}{\left(x+2\right)x^2}\right)$

$\mathrm{Minus}\left(\mathrm{R1}\,\mathrm{R2}\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{R3}≔\mathrm{Remainder}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\right)$

$\mathrm{R3}≔\mathrm{OrePoly}\left(\frac{5x^9+40x^8+151x^7+287x^6+302x^5+193x^4+58x^3-30x^2-32x-8}{x^3{\left(x+1\right)}^3{\left(x+2\right)}^4}\,-\frac{x^8+6x^7+13x^6+22x^5+23x^4+12x^3+x^2-8x-4}{{\left(x+1\right)}^2{\left(x+2\right)}^3{x}^2}\right)$

$R≔\mathrm{Remainder}\left['\mathrm{right}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\,'Q'\right)$

$R≔\mathrm{OrePoly}\left(\frac{x^2+4x+2}{x\left(x+1\right)\left(x+2\right)}\,-\frac{x^4-x^2-2x-1}{\left(x+2\right)x^2}\right)$

$Q$

$\mathrm{OrePoly}\left(-\frac{x^2+x+1}{x\left(x+1\right)\left(x+2\right)}\,\frac{x}{x+2}\right)$

$\mathrm{Minus}\left(\mathrm{Poly1}\,\mathrm{Add}\left(\mathrm{Multiply}\left(Q\,\mathrm{Poly2}\,A\right)\,R\right)\right)$

$\mathrm{OrePoly}\left(0\right)$

$R≔\mathrm{Remainder}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,A\,'Q'\right)$

$R≔\mathrm{OrePoly}\left(\frac{5x^9+40x^8+151x^7+287x^6+302x^5+193x^4+58x^3-30x^2-32x-8}{x^3{\left(x+1\right)}^3{\left(x+2\right)}^4}\,-\frac{x^8+6x^7+13x^6+22x^5+23x^4+12x^3+x^2-8x-4}{{\left(x+1\right)}^2{\left(x+2\right)}^3{x}^2}\right)$

$Q$

$\mathrm{OrePoly}\left(-\frac{4x^2+5x+2}{\left(x+1\right){\left(x+2\right)}^{2}x}\,\frac{x}{x+2}\right)$

$\mathrm{Minus}\left(\mathrm{Poly1}\,\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{Poly2}\,Q\,A\right)\,R\right)\right)$

$\mathrm{OrePoly}\left(0\right)$

$B≔\mathrm{SetOreRing}\left(x\,'\mathrm{shift}'\right)$

$B≔\mathrm{UnivariateOreRing}\left(x\,\mathrm{shift}\right)$

$\mathrm{Poly1}≔\mathrm{OrePoly}\left(\frac{1}{x}\,0\,\frac{x}{x+1}\,\frac{x^2}{x+2}\right)$

$\mathrm{Poly1}≔\mathrm{OrePoly}\left(\frac{1}{x}\,0\,\frac{x}{x+1}\,\frac{x^2}{x+2}\right)$

$\mathrm{Poly2}≔\mathrm{OrePoly}\left(x\,\frac{x+1}{x}\,x\right)$

$\mathrm{Poly2}≔\mathrm{OrePoly}\left(x\,\frac{x+1}{x}\,x\right)$

$Q≔\mathrm{Quotient}\left(\mathrm{Poly1}\,\mathrm{Poly2}\,B\right)$

$Q≔\mathrm{OrePoly}\left(\frac{1}{{\left(x+1\right)}^2}\,\frac{x^2}{\left(x+2\right)\left(x+1\right)}\right)$

$Q≔\mathrm{Quotient}\left['\mathrm{right}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,B\,'R'\right)$

$Q≔\mathrm{OrePoly}\left(\frac{1}{{\left(x+1\right)}^2}\,\frac{x^2}{\left(x+2\right)\left(x+1\right)}\right)$

$R$

$\mathrm{OrePoly}\left(\frac{2x+1}{x{\left(x+1\right)}^2}\,-\frac{x^4+x^3+x+2}{\left(x+2\right)x\left(x+1\right)}\right)$

$\mathrm{Minus}\left(\mathrm{Poly1}\,\mathrm{Add}\left(\mathrm{Multiply}\left(Q\,\mathrm{Poly2}\,B\right)\,R\right)\right)$

$\mathrm{OrePoly}\left(0\right)$

$Q≔\mathrm{Quotient}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,B\right)$

$Q≔\mathrm{OrePoly}\left(\frac{1}{\left(x-1\right){\left(x-2\right)}^2}\,\frac{x-2}{x}\right)$

$Q≔\mathrm{Quotient}\left['\mathrm{left}'\right]\left(\mathrm{Poly1}\,\mathrm{Poly2}\,B\,'R'\right)$

$Q≔\mathrm{OrePoly}\left(\frac{1}{\left(x-1\right){\left(x-2\right)}^2}\,\frac{x-2}{x}\right)$

$R$

$\mathrm{OrePoly}\left(\frac{x^3-6x^2+8x-4}{x\left(x-1\right){\left(x-2\right)}^2}\,-\frac{x^5-4x^4+5x^3-2x^2+x+1}{x^2{\left(x-1\right)}^2}\right)$

$\mathrm{Minus}\left(\mathrm{Poly1}\,\mathrm{Add}\left(\mathrm{Multiply}\left(\mathrm{Poly2}\,Q\,B\right)\,R\right)\right)$

$\mathrm{OrePoly}\left(0\right)$