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What is the algorithm for long division of polynomials with multiple . . . It seems to me that all the examples that illustrate how division isn't automatically well-defined for multivariate polynomials involve higher-order monomials If we restrict to polynomials where each monomial is multilinear, is there still a need to do all this Grobner basis business, ie defining an ordering etc?
abstract algebra - Proof of the polynomial division algorithm . . . The key idea of polynomial division is this: if the divisor has invertible lead coef $\,b\,$ (e g $\,b=1)\,$ 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 killing the leading term of
abstract algebra - Division algorithm for multivariate polynomials . . . 21 No, the polynomial division algorithm does not immediately generalize to multivariate rings Here is a simple proof Just as for $\rm\:\Bbb Z,\:$ a domain having an algorithm for division with smaller remainder, also enjoys Euclid's algorithm for gcds, which, in extended form, yields Bezout's identity
Why should I use synthetic division instead of long division of . . . 2 I know how to do synthetic division and long division However, I don't really see why I should bother remembering synthetic division since it can't be utilized in all scenarios Hence my question is: What type of problems require the usage of synthetic division to find an answer that long division would not allow me to find?
Pseudo-code for Polynomial Long Division - Mathematics Stack Exchange I'm busy writing a polynomial long division class in Java, and I see that Wikipedia provides a great example for performing the long division by hand However, when I compare it to the provided pseudo-code a few lines further on, something doesn't look quite right in the pseudo-code
Why is only the first (highest) term of the divisor in polynomial long . . . Polynomial long division is an algorithm By that I roughly mean that it is an entirely mechanical procedure that is guaranteed to finish in finitely many steps In order to ensure this, we want a notion of progress In polynomial long division, this progress is measured by the degree of the polynomial in the numerator So how to attain progress? All we need to do for that, is eliminate the