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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$\begingroup$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.
$\endgroup$ 8 $\begingroup$Purple math actually has a great explanation for what synthetic division is and how it works. You can find it here:
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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