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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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    Synthetic division and Horner's method for evaluating a polynomial are very intimately related. See [this discussion](http://books.google.com/books?hl=en&id=JPq0pS3wrx4C&pg=PA8), for instance.2012-07-15
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    [Synthetic division](http://en.wikipedia.org/wiki/Synthetic_division) is simply the [polynomial long division algorithm](http://en.wikipedia.org/wiki/Polynomial_division) *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.2012-07-15
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    @BillDubuque Thanks, that was perfect! Could you post your comment as an answer?2012-07-15
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    Khan academy has a great video on synthetic division. See: https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/dividing_polynomials/v/why-synthetic-division-works Great explanation!2013-09-15
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    It also seems that a year ago, I [wrote something](http://mixedmath.wordpress.com/2012/08/09/precalculus-supplement-synthetic-division/#more-619) for a precalculus student I had about synthetic division. I also mention this question - but never linked back.2013-09-16

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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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    ..... 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. :/2018-05-02