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# Prove properties about the zeros of a polynomial and its derivatives

Consider a polynomial . We say a number is a zero of multiplicity if

where .

1. Prove that if the polynomial has zeros in , then its derivative has at least zeros in . More generally, prove that the th derivative, has at least zeros in the interval.
2. Assume the th derivative has exactly zeros in the interval . What can we say about the number of zeros of in the interval?

1. Proof. Let denote the distinct zeros of in and their multiplicities, respectively. Thus, the total number of zeros is given by,

By the definition given in the problem, if is a zero of of multiplicity then

Taking the derivative (using the product rule), we have

Thus, again using the definition given in the problem, is a zero of of multiplicity .
Next, we know from the mean-value theorem for derivatives, that for distinct zeros and of there exists a number (assuming, without loss of generality, that ) such that . Hence, if has distinct zeros, then the mean value theorem guarantees numbers such that . Thus, has at least:

By induction then, the th derivative has at least zeros.

2. If the th derivative has exactly zeros in , then we can conclude that has at most zeros in .