Thread: Zero's of polynomial [Rolles Theorem]

1. Zero's of polynomial [Rolles Theorem]

Prove that if p(x) is a polynomial with n distinct zeros then p'(x) has at least n-1 zeros.

It seems that this is a fairly simply use of Rolle's theorem but I can't seem to see how to do it! Help would be great!

2. Originally Posted by nlews
Prove that if p(x) is a polynomial with n distinct zeros then p'(x) has at least n-1 zeros.

It seems that this is a fairly simply use of Rolle's theorem but I can't seem to see how to do it! Help would be great!
Here is an idea on how to get started. To simplify thing lets suppose that $\displaystyle p(x)$ has two roots lets call them $\displaystyle x_1,x_2$ and with out loss of generality we can have $\displaystyle x_1 < x_2$.

Rolle's theorem states that if $\displaystyle p(x_1)=p(x_2)=0$ then $\displaystyle p'(c)=0$ for some $\displaystyle c \in (x_1,x_2)$.

The above statement proves that if $\displaystyle p(x)$ has two roots then $\displaystyle p'(x)$ has $\displaystyle 2-1=1$ roots.

See if you can generalize this argument.