EuclideanDivision

EuclideanDivision[ expr1 / expr2, var ]

EuclideanDivision[ expr1 / expr2, var ] returns the Euclidean division of two polynomials, expr1 and expr2, with respect to var in algebraic form.

Example in out reduced
The Euclidean divison of x^3 - 1 and x^2 + 1 with respect to x. EuclideanDivision[ (x^3 - 1)/(x^2 + 1), x ] EuclideanDivision[ x, -x-1, x^2 + 1, x ] x + (-x - 1)/(x^2 + 1)