# Why does synthetic division work?

Synthethic division is commonly taught, but I have never actually had a proof/explanation shown to me. Why does it work?

Work So Far

I related the "$x$" to powers to 10, and then proceeded to relate synthetic division to non-polynomial division, but couldn't seem to find the correlation.

Research So Far

My teacher doesn't seem to have a valid explanation for why it works. A google search doesn't provide any good results either. All I seem to get is a Yahoo answers link with a badly formatted proof that makes it hard to understand and a physics forum link that links synthetic division to "normal division" by relating the "x" to 10, a conclusion I have already arrived at.

• Synthetic division and Horner's method for evaluating a polynomial are very intimately related. See this discussion, for instance. – J. M. is a poor mathematician Jul 15 '12 at 17:48
• Synthetic division is simply the polynomial long division algorithm optimized for a linear divisor. Said Wikipedia pages both do the same example. Put both pages side-by-side and it should be clear how the optimization works. – Bill Dubuque Jul 15 '12 at 17:50
• @BillDubuque Thanks, that was perfect! Could you post your comment as an answer? – user26649 Jul 15 '12 at 18:17
• Khan academy has a great video on synthetic division. See: khanacademy.org/math/algebra/multiplying-factoring-expression/… Great explanation! – user95045 Sep 15 '13 at 22:00
• It also seems that a year ago, I wrote something for a precalculus student I had about synthetic division. I also mention this question - but never linked back. – davidlowryduda Sep 16 '13 at 16:38

Per request, I post my comment here. Synthetic division is simply the polynomial long division algorithm optimized for the case when the divisor is linear (degree $1$). Said Wikipedia pages both do the same example. If you place these pages side-by-side and compare the associated steps then it should be clear how the optimization works.