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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.

3 Answers3

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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.

Bill Dubuque
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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. :/ – kitukwfyer May 02 '18 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. – KarmaPeasant Jan 28 '22 at 14:08
  • @user161005 See here for the idea behind the standard inductive proof. – Bill Dubuque Jan 28 '22 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 – KarmaPeasant Jan 28 '22 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? – Bill Dubuque Jan 28 '22 at 16:40
  • @BillDubuque I suspect, that it's because the proof was given in context of question, which in its turn based on content from abstract algebra book. In short, I don't understand the theorem from the linked question and what f, g, q, r are supposed to mean. Also, what are a,b,i,n and k in the proof itself? Also, it's unclear how "noninvertible lead coef" looks like – KarmaPeasant Jan 28 '22 at 16:55
  • @user161005 In that case it's probably best to ask a new question explaining your knowledge level so that you get answers at your level. – Bill Dubuque Jan 30 '22 at 19:57
  • @BillDubuque I already did it: https://math.stackexchange.com/questions/4369726/proof-request-polynomial-long-division-proof-aimed-at-pre-calculus-level-of-kno – KarmaPeasant Jan 30 '22 at 22:02
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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.

Marc Gauvin
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I urge you to look at the detailed answer I posted to a similar question here.

That question explains how to re-write the polynomial in nested form (Horner's form), which is a necessary first step to understanding how this algorithm works and also to appreciating its elegance and its efficiency. In essence, it's just repeated application of the Remainder Theorem, taken one step at a time.

Synthetic division is not presented very well in most high school math classes, which make it seem more mysterious (and simultaneously more mundane) than it actually is. It's based upon synthetic substitution, which is best to interpret as having the value for $x$ being substituted one degree at at time into the nested polynomial expression. At each step, the two leading terms are grouped and factored, so that the substitution reduces the degree by one.

The other key concept is the Remainder Theorem, which is true for polynomials of any degree, but is only being applied at each step in synthetic division to a linear factor of the form $(Ax+B)$. The theorem states that the remainder for $\frac{Ax+B}{x-a}$ is simply $Aa+B$. An important detail to note is that the quotient for the prior expression is simply $A$, because the divisor $x-a$ is monic (has a coefficient of $1$ on its leading term).

Put more simply: $Ax+B = A(x-a) + (Aa+B)$

The algorithm starts with the two leading terms and then proceeds, incorporating one new term at at time, by using the remainder from the previous step as the "A" for the next step. Each remainder becomes, in turn, the leading coefficient for the next step, which then becomes the coefficient of the corresponding quotient term, after division. Remainders are found at each step by substituting $a$ for the next $x$ in the Horner's form polynomial, according to the Remainder Theorem.

Michael Malione
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