Is there a software to long divide polynomials whose coefficients are variables?

Question

I want to divide the following polynomial (in terms of $t$) with coefficients in terms of $\lambda$.

$$(\lambda^6 - 5\lambda^4 + 6\lambda^2 - 1)t^5 + (\lambda^5 - 4\lambda^3 + 3\lambda^2)t^6$$

by $$ \lambda t^2 -\lambda^2 t + \lambda$$

The resulting quotient will include a fractional component (the numerator's degree will be strictly less than the denominator's degree -- is there any way to find the partial fraction decomposition of this expression, which will be in terms of $\lambda$).

EDIT:

I have not yet finished the hand-written calculation above, but this is what it might like:

$$ t(\frac{2\lambda^8 - 9 \lambda^6 + 2 \lambda^5 + 6 \lambda^4 - 4\lambda^2}{\lambda}) + t^3(\frac{2\lambda^6 - 9\lambda^4 + 3\lambda^3 + 6\lambda^2 -1 }{\lambda}) + \frac{t(\lambda^4 - 2\lambda) + (\lambda^3 - 4\lambda^2)}{\lambda t^2 - \lambda^2 t + \lambda}$$

Answer 1

In MAPLE:

n:=(lambda^6-5*lambda^4+6*lambda^3-1)*t^5+(lambda^5-4*lambda^3+3*lambda^2)*t^6;
d:=lambda*t^2-lambda^2*t+lambda;
convert(n/d,parfrac,t);

Answer 2

WolframAlpha can do polynomial division.

Quotient

Remainder

Partial Fractions

Note that in your case the partial fraction expansion doesn't give tons of extra information as the denominator is irreducible.

Answer 3

In Mathematica:

 PolynomialQuotientRemainder[(\[Lambda]^6 - 5 \[Lambda]^4 + 
 6 \[Lambda]^3 - 1) t^5 + (\[Lambda]^5 - 4 \[Lambda]^3 + 
 3 \[Lambda]^2) t^6, \[Lambda] t^2 - \[Lambda]^2 t + \[Lambda], t]

 {2 + 3 \[Lambda] - 5 \[Lambda]^2 - 21 \[Lambda]^3 + 23 \[Lambda]^4 + 
   9 \[Lambda]^5 - 14 \[Lambda]^6 + 2 \[Lambda]^8 + 
   t^4 (3 \[Lambda] - 4 \[Lambda]^2 + \[Lambda]^4) + 
   t^3 (-(1/\[Lambda]) + 9 \[Lambda]^2 - 9 \[Lambda]^3 + 2 \[Lambda]^5) + 
   t^2 (-1 - 3 \[Lambda] + 4 \[Lambda]^2 + 9 \[Lambda]^3 - 10 \[Lambda]^4 + 2\[Lambda]^6) + 
   t (1/\[Lambda] - \[Lambda] - 12 \[Lambda]^2 + 13 \[Lambda]^3 + 
   9 \[Lambda]^4 - 12 \[Lambda]^5 + 2 \[Lambda]^7), -2 \[Lambda] - 
   3 \[Lambda]^2 + 5 \[Lambda]^3 + 21 \[Lambda]^4 - 23 \[Lambda]^5 - 
   9 \[Lambda]^6 + 14 \[Lambda]^7 - 2 \[Lambda]^9 + 
   t (-1 + 3 \[Lambda]^2 + 15 \[Lambda]^3 - 18 \[Lambda]^4 - 
  30 \[Lambda]^5 + 35 \[Lambda]^6 + 9 \[Lambda]^7 - 
  16 \[Lambda]^8 + 2 \[Lambda]^10)}

In reality, with the Mathematica notebook front end, in input you would directly type the $\lambda$ and direct type the powers, and the output would have the same form:

$$\text{PolynomialQuotientRemainder}\left[\left(\lambda ^5-4 \lambda ^3+3 \lambda ^2\right) t^6+\left(\lambda ^6-5 \lambda ^4+6 \lambda ^3-1\right) t^5,\lambda +\lambda t^2-\lambda ^2 t,t\right]$$

$$\left\{2 \lambda ^8-14 \lambda ^6+9 \lambda ^5+23 \lambda ^4-21 \lambda ^3-5 \lambda ^2+3 \lambda +\left(\lambda ^4-4 \lambda ^2+3 \lambda \right) t^4+\left(2 \lambda ^5-9 \lambda ^3+9 \lambda ^2-\frac{1}{\lambda }\right) t^3+\left(2 \lambda ^6-10 \lambda ^4+9 \lambda ^3+4 \lambda ^2-3 \lambda -1\right) t^2+\left(2 \lambda ^7-12 \lambda ^5+9 \lambda ^4+13 \lambda ^3-12 \lambda ^2-\lambda +\frac{1}{\lambda }\right) t+2,-2 \lambda ^9+14 \lambda ^7-9 \lambda ^6-23 \lambda ^5+21 \lambda ^4+5 \lambda ^3-3 \lambda ^2-2 \lambda +\left(2 \lambda ^{10}-16 \lambda ^8+9 \lambda ^7+35 \lambda ^6-30 \lambda ^5-18 \lambda ^4+15 \lambda ^3+3 \lambda ^2-1\right) t\right\}$$