By using Rolle's Theorem, prove that:
if the real polynomial p of degree n has all its roots real, then so does its derivative p'.
give an example of a cubic polynomial p for which the converse fails.
let
be the n roots of a polynomial
Note: The derivative has degree n-1 so has at most n-1 roots
Consider the following by Rolle's theorem.
since
for i =1 to n-1
Then for each i There exists a
such that
These are each of the n-1 zeros of the derivative.
I guess this proof only works the zero's are not repeated(I think)
QED
I hope this helps
Good luck
I had an epiphany while playing some video games.
if it is a repeated root then the polynomial can be factored and written as
where f(x) is a polynomial
now we can take the derivative of p
so
So repeated real roots yield real roots of the derivative.
This can be generalized if a root is of multiplicty n the derivative will have a real zero of multiplicty n-1.
I hope this helps.