How does synthetic division work for increasing or decreasing the roots of a polynomial equation by a constant factor.

Question

I was browsing previous questions and found the logic behind synthetic division process was explained in

Why does synthetic division work? and also browsed the process behind in Purple math website, Wolfram Alpha. I can understand to a certain extent how it works when the divisor is of a polynomial of degree 1.

My question consists of two parts:

1. The logic behind the shortcut method is puzzling for synthetic division for a long polynomial division of degree more than 1 of the divisor, as explained in the link shared before in this question Wolfram Alpha part

2. The solution provided in the image to the question: Increase the roots of the equation x^4 - 16x^3 + 83x^2 +152*x + 84 = 0 by 4 and hence solve the equation; how it works through synthetic division( I can understand if I substitute y=x-4 the equation transforms into the desired result; but my question is how the diminishing of the roots by taking out the reminder part of synthetic division works together well through the iterative division method.

Answer 1

Once upon a time (around 1800) calculations were done only with paper and pencil and the help of limited mechanical devices (abacus, Babbage machine). At the same time, large scale scientific engineering required the mass production of numerical results, mostly linear algebra but also solutions of non-linear equations.

Thus the task presented itself to accurately compute roots of polynomials and power series for, e.g., 3 decimals after the dot and to then, if desired, improve this result by blocks of 3 digits with minimal effort (some of these tricks are used today in Kahan summation and other operations that extend floating point arithmetic without resorting to multi-precision types). Methods were proposed by people like Ruffini, Horner, Holdred and probably others and some form of teachable algorithm was devised and widely taught.

It seems that the basic block that most people took away from this as characteristic is the synthetic division, so the names of the inventors of the root-computing algorithms were, over time, attached to parts or interpretations of synthetic division. As far as I understand, the name Horner is mostly used for the efficient evaluation scheme for polynomials $$ p(x)=a_0+x(a_1+x(a_2+x(...+xa_n))). $$ Such a grouping of terms can already be found in Newton's work. From inside out this reads as $$ b_n=a_n\\ b_{n-1}=xb_n+a_{n-1}\\ ...\\ p(x)=b_0=xb_1+a_0 $$

The tabular scheme that is frequently used for the computation is also given the name Horner \begin{array}{c||c|c|c} a_k&a_n&a_{n-1}&...&a_0\\ \hline xb_{k+1}&*&xb_n&...&xb_1\\ \hline b_k&b_n&b_{n-1}&...&p(x)=b_0 \end{array}

The Ruffini theorem or rule is now that the numbers computed in the above scheme give exactly the result of synthetic division $$ p(y)=p(x)+(y-x)q(y)~~\text{ where }~~q(y)=b_ny^{n-1}+...+b_2y+b_1 $$ This result in itself was already known to antique-times mathematicians.

---

Now you can split $q$ by the same method as $$ q(y)=q(x)+(y-x)r(y)\implies p(y)=p(x)+(y-x)(q(x)+(y-x)r(y))) $$ and so on until finally you get $p(y)$ as polynomial $\tilde p$ in $(y-x)$. $$ p(y)=\tilde p(y-x)\iff p(y+x)=\tilde p(y). $$ If $y^*$ is a root of $p$, then $y^*-x$ is a root of $\tilde p$, so that the root set is shifted by $x$.

Answer 2

Courtesy from @LutzLehmann comments, I was able to find out the logic behind synthetic division aka Horner’s method in en.wikipedia.org/wiki/Horner%27s_method and I will accept it as the answer.