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.

Printable View

- May 16th 2008, 06:01 PMszpengchaoRolle's Theorem application
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. - May 16th 2008, 06:22 PMTheEmptySet
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 - May 16th 2008, 09:59 PMTheEmptySet
I had an epiphany while playing some video games. (Clapping)(Wink)

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. - May 17th 2008, 05:58 PMThePerfectHacker
You can find this here.