Who developed the mathematics of synthetic division?

Hi realblond-ga

Synthetic division was first described by Paolo Ruffini, an Italian
mathematician, in 1809.

http://en.wikipedia.org/wiki/Synthetic_division

Synthetic division is, in fact, also referred to as "division through
Ruffini's rule," and an example may be found here:

http://en.wikipedia.org/wiki/Polynomial_long_division#Synthetic_division.

Paolo Ruffini was also a philosopher and was licensed to practice
medicine. A short biography may be found here:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html

A bibliography of web references to Ruffini is here:

http://www.factbites.com/topics/Paolo-Ruffini

Hope this is helpful to you.

All the best, 

alanna-ga

Very helpful and insightful!