The Wikipedia article implies that Dan left out a factor of some power of the leading coefficient. I think I trust Dan on this one. On Tue, Jun 30, 2015 at 6:35 PM, Dan Asimov <dasimov@earthlink.net> wrote:
On Jun 30, 2015, at 3:05 PM, Henry Baker <hbaker1@pipeline.com> wrote:
I recall learning a bunch of algebra about the discriminant, which becomes zero when there are coincident roots.
https://en.wikipedia.org/wiki/Discriminant
Has someone come up with geometric insights about this particular formula ?
In the case of a quadratic, the formula is (x1-x2)^2, but this isn't the real number |x1-x2|^2. Perhaps the norm of the discriminant (DD*) is more important?
What about the discriminant of the cubic ? Shouldn't this say something interesting about the triangle in the complex plane formed by the roots?
The discriminant disc(P) of any polynomial P(x) in K[x] (K being some field) is the product of the squares of the differences between all pairs of roots.
(When K is a subfield of the reals, disc(P) is always real, because it's a symmetric polynomial Q(z_1,...,z_n) of the roots z_1,...,z_n of P(x), and it's a theorem that all symmetric polynomials are polynomials in the elementary symmetric functions . . . and the the elementary symmetric functions of the roots of P(x) are of course the coefficients.)
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