Talk:Hudde's rules

Rule 1 and the "modern theorem"
The definition of the polynomial $$a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n$$ numbers the coefficients just opposite to the way we define it at Polynomial and anywhere else throughout Wikipedia, which makes summation and derivation more cumbersome. Instead, I would just go with the common definition and write (if the expanded form with dots is even necessary):
 * $$f(x) = \sum_{k=0}^n a_k x^k = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 = 0$$

and similarly
 * $$\sum_{k=0}^n a_k b_k x^k = 0$$

This then lends itself, since the $$b_k$$ form an arithmetic progression, to the simplification
 * $$\sum_{k=0}^n a_k (c+k) x^k = 0$$
 * $$ (c+1) \sum_{k=0}^n a_k x^k + \sum_{k=0}^n a_k (k-1) x^k = 0$$
 * $$ f'(x) = 0$$

Thus showing the connection to the "modern theorem". But what's the name of the "modern theorem"; is it covered anywhere? (Maybe in its generalized form for any multiple roots.) ◄ Sebastian 21:05, 12 April 2018 (UTC)