macrosmath-modepackages
Is there in existence packages that recreate the following environments:
Some of the above have been discussed like in the How to present a vertical multiplication but what I am interested in is if there is a way to recreate the addition and division.
the xlop package does this sort of thing. It does warn that it uses "french conventions", but at least for multiplication it looks fine, to me.
disclaimer: i last did multiplication sums in school in the 1950s...
\documentclass{article}
\usepackage{xlop}
\begin{document}
\opmul{384}{56}\qquad
\end{document}
I'm afraid there isn't such a package to do calculations over finite fields. Arithmetic over finite fields GF(p^n) may be too complex for TeX. GF(2) and GF(p) are much easier, but there seems no such a package either.
To typeset long division manually, you can simply use an array. For example:
\documentclass{article}
\usepackage{amssymb}
\begin{document}
\[
x^3 - x + 1 = (x-1)(x^2+x) + 1 \in \mathbb{F}_3[x]
\]
\[
\renewcommand\arraystretch{1.2}
\begin{array}{*2r @{\hskip\arraycolsep}c@{\hskip\arraycolsep} *4r}
& && & 1 & 1 & 0 \\
\cline{3-7}
1 & -1 &\big)& 1 & 0 & -1 & 1 \\
& && 1 & -1 \\
\cline{4-6}
& && & 1 & -1 \\
& && & 1 & -1 \\
\cline{5-7}
& && & & & 1 \\
\end{array}
\]
\end{document}
If you need a lot of these long divisions, it is worth to write a program (not necessarily in TeX) to generate the code. It's not very difficult for polynomials over GF(p).
For edited question:
To increase the vertical space between array lines, you can redefine the factor \arraystretch as showed above. Or you can use makecell package to add a gap.
Best Answer
The addition is fairly simple and examples can be found at Numbering Vertical Addition Problems.
For the long division in your style you can use an
arrayenvironment.The MWE is shown below.