For the quotient of 2 polynomials polynomial division is carried out, showing all steps.
The remainder of the division is simplified in case of common linear factors of numerator and denominator by shortening.
If the numerator polynomial has at least the same degree as the denominator polynomial,
then you get a polynomial as a result and in general, i.e. if the division does not work out exactly, a remainder of the division.
The remainder of the division, if any, then is a proper rational function. For clarity it's diplayed in a different color.
An nth degree polynomial is defined here via its n+1 coefficients. Coefficients of course also can be 0.
Alternatively, the polynomial can also be entered directly as the sum of powers of x (with ^ as power operator).
For example, there are 2 ways to define the same 5th degree polynomial:
x^5 - 8x^3 + 2x + 1 or 1 0 -8 0 2 1