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I'm still researching, but, at the moment, I would give credit to Rene Descartes who noted the relationship between the rational roots of a polynomial and the coefficients of the first and last terms. See Google Books link below (bottom of page 243 of text.
http://books.google.com/books?id=5NIDb84jOmcC&pg=PA243&lpg=PA243&dq=history+rational+roots+theorem&source=bl&ots=KYMZS2m4RE&sig=WZS7COaip1KcW4ZfLSHBgcp-Co8&hl=en&ei=3sMmStOACJqutAPFvMHFBg&sa=X&oi=book_result&ct=result&resnum=10#PPA243,M1
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