Talk:Hudde's rules

Rule 1 and the "modern theorem"

The definition of the polynomial $a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}$ numbers the coefficients just opposite to the way we define it at Polynomial#Definition 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):

$f(x)=\sum _{k=0}^{n}a_{k}x^{k}=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=0$

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)