Andrew, Read up on resultants:
sage: R.<a1,a2,b1,b2,x,y>=QQ[] sage: f=x^2+a1*x+a2 sage: g=x^2+b1*x+b2 sage: f(x=y).resultant(g(x=x-y),variable=y) a1^2*b1*x + a1*b1^2*x + a1^2*x^2 + 3*a1*b1*x^2 + b1^2*x^2 + 2*a1*x^3 + 2*b1*x^3 + x^4 + a1*a2*b1 + a2*b1^2 + a1^2*b2 + a1*b1*b2 + 2*a1*a2*x + 2*a2*b1*x + 2*a1*b2*x + 2*b1*b2*x + 2*a2*x^2 + 2*b2*x^2 + a2^2 - 2*a2*b2 + b2^2 sage: f(x=y).resultant(g(x=x+y),variable=y) a1^2*b1*x - a1*b1^2*x + a1^2*x^2 - 3*a1*b1*x^2 + b1^2*x^2 - 2*a1*x^3 + 2*b1*x^3 + x^4 - a1*a2*b1 + a2*b1^2 + a1^2*b2 - a1*b1*b2 - 2*a1*a2*x + 2*a2*b1*x - 2*a1*b2*x + 2*b1*b2*x + 2*a2*x^2 + 2*b2*x^2 + a2^2 - 2*a2*b2 + b2^2 The last two results are what you want (roots alpha+beta, resp. alpha- beta). John Cremona On Oct 24, 6:23 pm, andrew ewart <aewartma...@googlemail.com> wrote: > is there a way of doing what uve done except considering the pair of > polynomials > x^2+a1*x+a0 > x^2+b1*x+b0 > where a0,a1,b0,b1 lie in QQ but their exact values r not known > and al is a root of the first and be is a root of the second? > > > > > > > > On Sun, Oct 24, 2010 at 5:22 PM, Yann <yannlaiglecha...@gmail.com> wrote: > > On Oct 23, 9:39 pm, andrew ewart <aewartma...@googlemail.com> wrote: > > > if alpha is a root of x^2+a_1*x+a_0 and beta is a root of x^2+b_1*x > > > +b_0 (both polynomials r in QQ), then how do i construct code such > > > that it can tell me the minimum polynomials of the following roots > > > alpha+beta > > > and > > > alpha*beta > > > Hi, I'm not sure if I understood your question, but this might be what > > you want: > > > sage: R.<x> = QQbar[] > > sage: p = 1/7*x^2-x+4 > > sage: q = x^2+x+1/3 > > sage: p.roots() > > [(3.500000000000000? - 3.968626966596886?*I, 1), (3.500000000000000? + > > 3.968626966596886?*I, 1)] > > sage: a = p.roots()[0][0] > > sage: b = q.roots()[0][0] > > sage: (a+b).minpoly() > > x^4 - 12*x^3 + 257/3*x^2 - 298*x + 5503/9 > > sage: (a*b).minpoly() > > x^4 + 7*x^3 + 77/3*x^2 + 196/3*x + 784/9 > > > I hope this helps. > > > Yann > > > -- > > To post to this group, send email to sage-support@googlegroups.com > > To unsubscribe from this group, send email to > > sage-support+unsubscr...@googlegroups.com<sage-support%2bunsubscr...@google > > groups.com> > > For more options, visit this group at > >http://groups.google.com/group/sage-support > > URL:http://www.sagemath.org -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
