This will be a long post and there will be a TL;DR at the end.
I've recently been re-reading topics on polynomial division to brush up my knowledge on them but sometimes I get a little confused and conflate some of the terms because they seem to mean similar things and I'e seen some people online use them interchangeably so I wanted to ask on the differences between the following terms and whether or not my understanding and summary of them is correct.
• Euclidean Division of Polynomials:
Essentially, the polynomial analogue to Euclidean Division of Integers but instead of $$ a = bq + r \tag{$a, b, q, r \in \mathbb{Z}\,$, $b\neq 0\,$, $0 \leq r < |b|$,} $$ where $a$ is the dividend, $b$ is the divisor, $q$ is the quotient, and $r$ is the remainder,
we have $$ f(x) = g(x)Q(x) + R(x) \tag{$g(x) \neq 0\,$, either $R = 0$ or $\deg(R) < \deg(g)$,} $$ where $f(x)$ is the dividend polynomial, $g(x)$ is the divisor polynomial, $Q(x)$ is the quotient polynomial, and $R(x)$ is the remainder polynomial.
• Polynomial Long Division:
Polynomial long division is an algorithm that implements the Euclidean division of polynomials for dividing a polynomial by another polynomial of the same or lower degree similar to the arithmetic technique of Long Division in which the variable $x$ is replaced (in base $10$) by the specific number $10$.
Examples:
$\mathbf{1.}$ The answer to "Find the quotient and the remainder of the polynomial division of $x^3 - 2x^2 - 4$, the dividend, by $x-3$, the divisor." would be the following.
The polynomial above the bar is the quotient $Q(x)$, and the number left over, $5$, is the remainder $R$.
Hence the quotient and remainder are: $$ \begin{align*} Q(x) &= x^2 + x + 3, \\ R &= 5. \end{align*} $$
$\mathbf{2.}$ The answer to "Find the quotient and the remainder of the polynomial division of $x^3 - 12x^2 - 42$, the dividend, by $x^2 + x - 3$, the divisor." would be the following.
Hence the quotient and remainder are: $$ \begin{align*} Q(x) &= x - 13, \\ R(x) &= 16x - 81. \end{align*} $$
$\mathbf{3.}$ The answer to "Find the quotient and the remainder of the polynomial division of $6x^3 + 5x^2 - 7$, the dividend, by $3x^2 - 2x - 1$, the divisor." would be the following.
Hence the quotient and remainder are: $$ \begin{align*} Q(x) &= 2x + 3, \\ R(x) &= 8x - 4. \end{align*} $$
• Polynomial Synthetic Division:
Polynomial synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division.
Examples:
$\mathbf{1.}$ The answer to "Find the quotient and the remainder of the polynomial division of $x^3 - 2x^2 - 4$, the dividend, by $x-3$ (whose root is $x = 3$), the divisor." would be the following.
Hence the quotient and remainder are: $$ \begin{align*} Q(x) &= x^2 + x + 3, \\ R &= 5. \end{align*} $$
$\mathbf{2.}$ The answer to "Find the quotient and the remainder of the polynomial division of $x^3 - 12x^2 - 42$, the dividend, by $x^2 + x - 3$, the divisor." would be the following.
Negate the coefficients of the divisor and write in every coefficient but the first one on the left in an upward right diagonal.
Count the terms to the left of the bar. Since there are two, the remainder has degree one ($\deg(R) < \deg(g)$) and this is the two right-most terms under the bar.
Hence the quotient and remainder are: $$ \begin{align*} Q(x) &= x - 13, \\ R(x) &= 16x - 81. \end{align*} $$
$\mathbf{3.}$ The answer to "Find the quotient and the remainder of the polynomial division of $6x^3 + 5x^2 - 7$, the dividend, by $3x^2 - 2x - 1$, the divisor." would be the following.
Negate the coefficients of the divisor and write in every coefficient but the first one on the left in an upward right diagonal.
Note the extra row at the bottom. This is used to write values found by dividing the "dropped" values by the leading coefficient of $g(x)$ (in this case, indicated by the $\div\,3$; note that, unlike the rest of the coefficients of $g(x)$, the sign of this number is not changed).
Count the terms to the left of the bar. Since there are two, the remainder has degree one ($\deg(R) < \deg(g)$) and this is the two right-most terms under the bar (the values of the remainder are not divided by the leading coefficient of the divisor).
Hence the quotient and remainder are: $$ \begin{align*} Q(x) &= 2x + 3, \\ R(x) &= 8x - 4. \end{align*} $$
• Gauss's Lemma for Polynomials:
The primitive content of a polynomial $f(x) = a_0 + a_1 x + \cdots + a_n x^n$ is defined to be $\gcd(a_0, a_1, \ldots, a_n)$. The primitive part of such a polynomial is the quotient of the polynomial by its content, i.e., $\dfrac{f(x)}{\gcd(a_0, a_1, \ldots, a_n)}$. A polynomial is primitive if its content equals $1$. Thus the primitive part of a polynomial is a primitive polynomial.
Gauss's lemma asserts that the product of two primitive polynomials is primitive, i.e, for $f(x) = a_0 + a_1 x + \cdots + a_n x^n$, $g(x) = b_0 + b_1 x + \cdots + b_m x^m$, and $f(x)g(x) = c_0 + c_1 x + \cdots + c_{m+n} x^{m+n}$, we have
$$ \gcd(c_0, c_1, \ldots, c_{m+n}) = 1 \iff (\gcd(a_0, a_1, \ldots, a_n) = 1) \land (\gcd(b_0, b_1, \ldots, b_m) = 1). $$
I still don't understand the relation to the following Rational Root Theorem even after reading the proof.
• Rational Root Theorem
The rational root theorem, a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials, states that for each rational solution $x = \frac{p}{q}$ of a polynomial equation $$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0$$ (with integer coefficients $a_i \in \mathbb{Z}$ and $a_0, a_n \neq 0$), written in lowest terms so that $p$ and $q$ are relatively prime, satisfies:
$1.$ $p$ is an integer factor of $a_0$.
$2.$ $q$ is an integer factor of $a_n$.
Example:
Every rational root of the polynomial $$ 3x^3 - 5x^2 + 5x - 2 $$ must be among the numbers symbolically indicated by: $$ \pm\frac{1,2}{1,3} = \pm\left\{1, 2, \frac{1}{3}, \frac{2}{3}\right\} $$ These $8$ root candidates $x = r$ can be tested by evaluating $P(r)$
I still don't understand the justification of the efficient process above (and is it generalized?)
• Polynomial Remainder Theorem (or little Bézout's theorem)
The polynomial remainder theorem or little Bézout's theorem states that for the Euclidean division of polynomials form
$$ f(x) = g(x)Q(x) + R(x) = (x - r)Q(x) + R,$$
the remainder, $R$, of the division of the polynomial $f(x)$ by a linear polynomial $g(x) = x - r$ is equal to $f(r)$.
• Factor Theorem
Factor theorem is a special case of the polynomial remainder theorem (when $R = 0$) which states $$ f(x) \text{ has a factor } (x - r) \iff f(r) = 0. \tag{i.e., $r$ is a root} $$
• Horner's Method (or Horner's Scheme)
Horner's method or Horner's scheme is an algorithm for polynomial evaluation based on Horner's rule: $$ a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-1} x^{n-1} + a_n x^n = a_0 + x\left( a_1 + x\left( a_2 + \cdots + x\left(a_{n-1} + a_n\right) \right) \right) $$ and essentially uses polynomial synthetic division (divide $f(x)$ by $g(x) = x - x_0$) and applies the polynomial remainder theorem ($f(x_0) = R$).
Alternatively, Horner's method also refers to a method for approximating the roots of polynomials, described by Horner in 1819. It is a variant of the Newton–Raphson method made more efficient for hand calculation by the application of Horner's rule.
• Ruffini's Rule
Ruffini's rule is a method for computation of the Euclidean division of a polynomial and is a special case of polynomial synthetic division in which the divisor is a linear factor $(x - r)$.
• TL;DR
$1.$ Euclidean Division of Polynomials: $f(x) = g(x) Q(x) + R(x)$.
$2.$ Polynomial Long Division: Long division but for polynomials.
$3.$ Polynomial Synthetic Division: The division part of Horner's method or Horner's scheme.
$4.$ Gauss's Lemma for Polynomials: If $f(x)$ and $g(x)$ are primitive polynomials over the integers, their product $f(x)g(x)$ is also primitive.
$5.$ Rational Root Theorem: Each possible rational solution of $f(x) = a_0 + a_1 x + \cdots + a_n x^n = 0$ is $x = \frac{p}{q}$ where $\gcd(p,q) = 1$ and $p \mid a_0 \;\land\; q \mid a_n$.
$6.$ Polynomial Remainder Theorem: $f(x) = (x - r) Q(x) + R \iff f(r) = R$.
$7.$ Factor Theorem: $f(x) = (x - r) Q(x) \iff f(r) = 0$.
$8.$ Horner's Method (or Horner's Scheme):
a. Polynomial evaluation (synthetic division + polynomial remainder theorem)
b. Polynomial root-finding (initial educated guess of $x_0$ where $f(x_0) = 0$ + Newton–Raphson method + Horner's method + iteration)
$9.$ Ruffini's Rule: A special case of synthetic division in which the divisor is a linear factor $(x - r)$.
Are my understanding of each of them accurate?
Thank you if you've managed to read this far and I would be grateful for any answers I might receive.