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Try using polynomials a polynomial ring.

Then you can define

• two polynomials in it,
• the rational fraction obtained by dividing one by the other
• the quotient and remainder

Define a polynomial ring:

sage: R.<x> = QQ[]

Or just a polynomial ring generator:

sage: x = polygen(QQ)

Define two polynomials:

sage: a = 3*x^3 + x^2 - 3*x + 5
sage: b = x + 1

Divide:

sage: c = a / b
sage: c
3*x^3 + x^2 - 3*x + 5)/(x + 1)

Quotient and remainder

sage: q = a // b
sage: q
3*x^2 - 2*x - 1

sage: r = a % b
sage: r
6

Both at once

sage: q, r = a.quo_rem(b)
sage: q
3*x^2 - 2*x - 1
sage: r
6

Check:

sage: q + r/b
(3*x^3 + x^2 - 3*x + 5)/(x + 1)