| dbp:proof
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- By the fundamental theorem of algebra, is a product of linear factors as
:
where the complex numbers are the – not necessarily distinct – zeros of the polynomial , the complex number is the leading coefficient of and is the degree of .
For any root of , if it is also a root of , then the theorem is trivially true. Otherwise, we have for the logarithmic derivative
:
Hence
:.
Taking their conjugates, and dividing, we obtain as a convex sum of the roots of :
: (en)
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