On 5/19/10, mcintosh@servidor.unam.mx <mcintosh@servidor.unam.mx> wrote:
Quoting Fred lunnon <fred.lunnon@gmail.com>: ... You wouldn't think that such a simple formula would cause such trouble, but I finally understood the source when I graphed the roots as functions of the coefficients - or rather the coefficients as functions of the roots. A family of straight lines intersects a family of hyperbolas, trouble arising near the case of equal roots where the intersections are tangencies.
Such a picture would be a nice adjunct to a numerical analysis book - for those who like to see pictures illustrating their mathematics.
I have never seen this done in a textbook. But Bob Churchhouse in his Numerical Analysis course at Cardiff always used to discuss the behaviour of the coefficients as functions of the roots. I thought it a most useful exercise, and did not hesitate to appropriate it for my own course when the time came. WFL