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Math Help - Zero's of polynomial [Rolles Theorem]

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    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!
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    Quote Originally Posted by nlews View Post
    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 p(x) has two roots lets call them x_1,x_2 and with out loss of generality we can have x_1 < x_2.

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

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

    See if you can generalize this argument.
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