> Neither Maple nor Mathematica nor MuPad can factor your polynomial.
Can they Solve x^5+20*x+32 in closed form?
> What I meant was that "solve"...seems to be more powerful than
> "factor" even though both commands do essentially the same thing.
> That is the same thing Maple does. I simply wondered, why.
Factor is *defined* as the rational factorization, so doesn't introduce
radicals. Besides not using the quadratic formula etc., there are other
"factorizations" which it doesn't produce (but Solve does), e.g. x^7-1 =
(x-1)*(x-exp(2*%i*%pi/7))*....
Big complicated expressions in nested radicals are usually not terribly
useful. (Granted, they *occasionally* are -- especially when they
simplify nicely.) Having something in the form
(x-verybigmess1)*(x-verybigmess2)*(x-verybigmess3) rarely gives useful
insight, especially since simplifying nested radicals is difficult (and
Maxima doesn't really do it much).
Even in very simple cases, it's not clear what insight you get from
introducing radicals. Which is "better factored":
y^2+x*y+x^2
or
(SQRT(3)*%I-1)*(2*y+(1-SQRT(3)*%I)*x)*((SQRT(3)*%I+1)*y+(SQRT(3)*%I-1)*x
)/8
Compare the situation with integration. Do you prefer
integrate(1/(x^2+1),x) => atan(x) or => %i*log(x+%i)/2-%i*log(x-%i)/2?
They are mathematically equivalent....
> > Would you be willing to implement it for Maxima?
> If I did it, if I started to temper with Maxima, the number of Maxima
> users would rapidly decrease... Maybe I should ask Wofram research to
pay
> me for that...
Be a little daring!: try working on it. The worst that can happen is
that the result isn't good enough to include in Maxima -- but you will
probably have learned a lot. The best that can happen is that you've
increased the power of Maxima, and gained fame and fortune (well, maybe
a little fame, but not any fortune...).
-s