Does Descartes Rule of Signs Count Multiplicities in Its Upper Bound for Roots?

[wumple]

Hi,

Does the Descartes rule of signs count multiplicities when giving its upper bound for roots? That is if I have 3 sign changes, does that mean there is a maximum of 3 positive roots counting multiplicities or not counting multiplicities?

Thanks

 

[Infinitum]

Each root, even if the same, is counted separately.

 

[HallsofIvy]

For example, $(x- 2)^3= x^3- 6x^2+ 12x- 8= 0$ has three sign changes and three positive roots- counting "2" as a triple root.

Actually this isn't a very good example because Descarte's rule of signs does not actually "count" roots. DesCarte's rule of signs says only that the number of positive roots is at most equal to the number of sign changes or is less by a multiple of two. Here, Descarte's rule of signs says that the number of postive roots is either 3 or 1 so it is not clear if it "counting" multiple roots.

But the number of sign changes of $(x- 2)^2= x^2- 4x+ 4= 0$ is 2 so Descarte's rule of signs say the number of positive roots is 2 or 0. And it clearly is not 0.