Ruffini’s rule, synthetic division, Horner’s form etc

It seems there is a great confusion of terms in this area.
What is commonly called synthetic division is just one type of synthetic division, division of a polynomial by x-r, for which it seems the best name is Ruffini’s rule.
Horner’s rule, or Horner’s form, is a polynomial in the form:

    \[(((a_n x+a_{n-1})x+a_{n-2})x+\dots+a_1)x+a_0\]

Horner’s method is a way of finding roots of a polynomial, by repeatedly reducing the equation by the integer part of a root, and multiplying the coefficients by factors of 10 to obtain further digits.

These are related but separate things.

E. J. Barbeau’s Polynomials – a problem book describes evaluating a polynomial by ‘constructing Horner’s table’ (i.e. plugging in e.g. x=3 to the Horner form) as ‘Horner’s method’. And then shows that this is an identical process to dividing the polynomial by (x-3). I suppose the confusion comes from these 2 procedures being identical. Although neither is apparently Horner’s method. He does division by polynomials of a higher degree with “Horner’s Method of Synthetic Division which can be regarded as a generalization of his method for division by a binomial (x-c)“.
++++(He also adapts Horner’s table to convert powers and polynomials to falling (factorial) powers and polynomials – see below.)

Dividing a polynomial P(x) by x-a with Ruffini’s rule either results in a remainder of 0, in which case a is a root, or a remainder R.
The polynomial remainder theorem states:
If a polynomial P(x) is divided by x-a, then the remainder is a constant given by P(a).

    \begin{align*} \frac{P(x)}{x-a}&=Q(x)+\frac{R}{x-a}\\ P(x)&=Q(x)(x-a)+R\\ \text{So } \quad P(a)&=Q(a)(a-a)+R\\ P(a)&=R  \end{align*}

    \begin{align*} \text{e.g.} \qquad P(x)&=2x^3+3x^2+4x+5\\ \frac{P(x)}{x-1}&=2x^2+5x+9+\frac{14}{x-1}\\ P(x)&=(2x^2+5x+9)(x-1)+14\\ P(1)&=(..)(1-1)+14\\ &=14 \end{align*}

Barbeau’s table used to convert r^4 to falling powers:

Mathews, Walker – Mathematical Methods of Physics, 1971, p361-3

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