Oh, never mind then. I have a feeling that I probably proved this at one point, but I don't remember how.
Wait, I might have something. If you express a polynomial with multiple roots as product of several terms of (x-c)n
, where c is the root and n is the multiplicity, like x2
, you can just write out the general form of such a polynomial. Then you can take the derivative, and show that it will always have the same roots, because the derivative will contain the term n(x-c)n-1
multiplied by a bunch of other stuff. It might get a little messy, because of the product rule, but it's doable.
I hope you're working with real roots only?