Why the Euclidean division of polynomials way actually works?

What is the deep insight behind it?

Thank you in advance.

  • $\begingroup$ Suppose we are working over a field (say the reals, or rationals). We are attempting to divide $A(x)$ by $P(x)$. If the degree of $A$ is equal or bigger than the degree of $P$, then by subtracting a suitable multiple of $P(x)$ we can always lower the degree of $A$ by at least $1$. So the process must terminate. $\endgroup$ – André Nicolas Mar 16 '14 at 17:37
  • $\begingroup$ Check Wikipedia's page on Euclidean domains. The ring of polynomials $\mathbb R[x]$ is a Euclidean domain with as Euclidean function $f(p(x))$ the degree of the polynomial $p$. The Euclidean algorithm works on Euclidean domains. $\endgroup$ – lvaneesbeeck Mar 16 '14 at 17:38
  • $\begingroup$ Huh? What's that field, ring stuff? $\endgroup$ – user135823 Mar 16 '14 at 17:56

The key idea of the polynomial division algorithm is this: if the leading coefficient of the divisor $= 1$ (or is invertible), and the dividend has degree $\ge$ the divisor, then we can $\rm\color{#c00}{scale}$ the divisor so that it has the same degree and leading coef as the dividend, then subtract it from the dividend, thereby canceling the leading term of the dividend; then recursively apply this process to the remaining part of the dividend, which has smaller degree (since we killed the leading term of the dividend), viz.

$$ (\overbrace{a x^{k+n} + f}^{\rm dividend}) - \color{#c00}{a x^k} (\overbrace{x^n + g}^{\rm divisor})\ =\ f-ax^kg$$

$$\ \Rightarrow\ \dfrac{a x^{k+n}+f}{x^n+g}\, =\ \color{#c00}{a x^k} +\!\!\! \underbrace{\dfrac{f-ax^k g}{x^n + g}}_{\large\rm recurse\ on\ this}$$

where the second equation arises from the first by dividing through by $\,x^n + g.\,$ The long division algorithm for polynomials is simply a convenient tabular arrangement of the process obtained by iterating this descent process till one reaches a dividend having smaller degree than the divisor.

Remark $\ $ If one seeks a deeper understanding one can view this as a special case of multivariate polynomial division algorithms, such as the Gröbner basis algorithm. One gains further insight from this more general perspective on the descent process, e.g. in terms of monomial orderings.


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