I am a student and I definitely agree with Chris Seberino here. I don't think it is the job of software (or, rather, it is not good CAS design) to try and teach people mathematics in this way. I think that the CAS should be as convenient and intuitive as possible. As an example, I didn't even know that the functionality with rings that projetmbc highlights in his post was even possible, I simply assumed that Sage was limited in how it could do factorization. This is coming from someone who, in comparison to the rest of the math students at his school, is much more willing to read the documentation and fiddle with the function and google for solutions than most others, and I still didn't find this out till just now (and not for lack of searching on previous occasions, I should say). Also, w/ respect to Dima's statement about getting used to domains, I do not think that is exactly related here. The "Symbolic ring" which has the property of defaulting to the most expanded form of an expression is not a mathematical concept that I have ever heard of, it is behaviour that is part of Sage. It is behaviour that is very disconcerting for a new user using the factor function. I should add, on what I believe to be an unrelated note, that even if we were debating whether factorization should default to the reals or the complex numbers or something similar, I would argue that it should default to the most widely utilized behaviour, with the other options being left as just that, options (that you can turn on by keyword). So in the case of factoring to reals or complex roots, I would say that it should default to real roots, with a keyword allowing complex roots. Back to the main point, I do wish the factor function was made more intuitive, and that factoring in SR would recognize when it is integers and actually do something instead of giving you back what you put in.
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