# 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. Jul 15, 2012 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. Jul 15, 2012 at 17:50
• @BillDubuque Thanks, that was perfect! Could you post your comment as an answer?
– user26649
Jul 15, 2012 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, 2013 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. Sep 16, 2013 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.

• ..... I don't know if the examples changed or if I'm just dumb. Six years in the future here and I am sadly confused. :/ May 2, 2018 at 23:25
• This could work as a proof if we proved polynomial long division beforehand. Which doesn't seem to be the case to me. Jan 28 at 14:08
• @user161005 See here for the idea behind the standard inductive proof. Jan 28 at 14:55
• @BillDubuque What knowledge am I supposed to have in ordert to understand the proof that you linked? I try to understand it, but it feels like I'm missing something. My current knowledge of math is currently about school level pre-calculus Jan 28 at 16:28
• @user161005 It was tagged abstract-algebra so written at that level. But it doesn't really use any abstract algebra. I suggest you first concentrate on the special case $\,b = 1\,$ of the Key Idea (which uses only simple polynomial arithmetic). Where are you having difficulty? Jan 28 at 16:40

Purple math actually has a great explanation for what synthetic division is and how it works. You can find it here: https://www.purplemath.com/modules/synthdiv.htm

Basically the explanation is the fact that we use synthetic division to find factors of polynomials, which essentially is what division is. If the remainder of synthetic division is zero, then the divisor is a factor. The important thing here is that synthetic division only divides a polynomial by a linear factor.

I can understand the confusion. We use AVP matchbooks for precalculus math 12, and while the books are otherwise great, the explanation for synthetic division is sadly lacking.