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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.

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  • $\begingroup$ Synthetic division and Horner's method for evaluating a polynomial are very intimately related. See this discussion, for instance. $\endgroup$ – J. M. is a poor mathematician Jul 15 '12 at 17:48
  • $\begingroup$ 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. $\endgroup$ – Bill Dubuque Jul 15 '12 at 17:50
  • $\begingroup$ @BillDubuque Thanks, that was perfect! Could you post your comment as an answer? $\endgroup$ – user26649 Jul 15 '12 at 18:17
  • $\begingroup$ Khan academy has a great video on synthetic division. See: khanacademy.org/math/algebra/multiplying-factoring-expression/… Great explanation! $\endgroup$ – user95045 Sep 15 '13 at 22:00
  • $\begingroup$ 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. $\endgroup$ – davidlowryduda Sep 16 '13 at 16:38
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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.

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    $\begingroup$ ..... 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. :/ $\endgroup$ – kitukwfyer May 2 '18 at 23:25
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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.

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