Powers of irreducible polynomials
I'm working on a WeBWorK project to demonstrate the relatively new WeBWorK-Sage connectivity. We'd like to demonstrate some math problems that one could program in WeBWorK using Sage more cleanly than without Sage.
To that end I'm trying the following: WeBWorK would display a randomly generated polynomial in Z[x]. Let's say it could be degree 2, 3, or 4 with integer coefficients from [-10,10]. A student has to factor it completely.
I'm able to send the student's answer to Sage as a string, properly formatted for Sage. It might be '(x^2+1)(2x+1)' or '2(x^2+3x+2)*(x+2)^2'. But how can I evaluate whether or not the answer is fully factored? (For the moment, I'm not concerned about factoring out integers.)
So far, I have done this, where poly is the string passed to Sage:
factored = True if (poly).operator()!=operator.mul: if (poly).polynomial(base_ring=QQ).is_irreducible(): factored = True else: factored = False else: factors = (poly).operands() for factor in factors: factored = factored and factor.polynomial(base_ring=QQ).is_irreducible() print factored
And factored is True if all of the factors in poly were irreducible, and False otherwise. This is not what I want since I want something like '(x+1)^2*(x+2)' to count as factored. Is there a quick test for whether or not a polynomial is a power of an irreducible? Or a constant multiple of such a thing?