Paolo Ruffini and Synthetic Division.

I love synthetic division.  I love it so much but unfortunately, it doesn't work on anything other than binomial factors. I love it's beauty and simplicity but I've often wondered about it's history.  Who provided the first consolidation of that knowledge?

From what I can tell, the creation of synthetic division is attributed to Paolo Ruffini in 1804.  He is an Italian Mathematician who lived from 1765 to 1822.  The work he did was a forerunner of the Algebraic Theory of Groups and he tried to show there is no solution to a quintic equation without radicals.

In addition to to being a mathematician, he was also a physician and philosopher.  Unfortunately after Napoleon took over, Ruffini lost his teaching position because he refused to take an oath of allegiance but that did not stop him.  He worked as a doctor while continuing his mathematical research but once Napoleon was defeated, Ruffini returned to his university position.

What we refer to as synthetic division in high school is also known as Ruffini's rule which allows people to divide a polynomial by a linear factor to find the zero's of the equation.  As you know, if there is no remainder, you have found a zero but if there is a remainder, you have found the y value of the equation at a certain x value .  It is also much more efficient than using algebraic long division to solve the same problem.

Around 1800 or so, the Italian Scientific Society of Forty opened a competition asking people to provide a method that could be used to find any roots of a polynomial.  In the end, the society received five entries but Ruffini's was declared the winner in 1804 and his paper was published as part of the award.  He refined and republished the paper in 1809 and 1813.

We have students practice it on paper by following certain steps but emathlab  has a really great practice section to help students learn the process.  If you get anything wrong, it will correct the work on the problem so you can see where you went wrong.

In addition, this article looks at ways to extend the use of synthetic division with modifications for x^2 - a and x^2 - bx + c problems.  I had never seen these before because I'd been told you cannot use synthetic division for anything other than linear factors.  I had trouble with the x^2 - a one but not the last one as it made perfect sense.  This site has a wonderful pdf with problems, explanations, a bit of history on synthetic division.

I've always wondered who came up with synthetic division because when I next teach it, I can include a bit of the history so that students will know more about it.  Let me know what you think, I'd love to hear. Have a great day.