The Polynom Package

Abstract

The polynom package implements macros for manipulating polynomials,
for example it can typeset long polynomial divisions. Such long divisions can
be shown step by step. The main test case and application is the polynomial
ring in one variable with rational coefficients.

1 Introduction

Donald Arseneau has contributed a lot of packages to the community. In
particular, he posted macros for long division on comp. text. tex , which were
also published in the TUGboat [1]. With these definitions, one could just write
\longdiv{12345}{13} to get the long division shown in Figure 1 (a). In fact, that

(a) \longdiv{12345}{13}	(b) \polylongdiv {X^3+X^2-1}{X-1}
Figure 1: Integer and polynomial long division

integer long division has been typeset using the code from the location cited. The
polynom package allows to do the similar job with polynomials, see Figure 1 (b).
Figure 2 shows partial long divisions.

An application of polynomial division is shown in Figure 3: the Euclidean
algorithm to determine a greatest common divisor of two polynomials. Note that
in the case here, a greatest common divisor is uniquely determined up to a scalar
factor, so X - 1 and are both greatest common divisors in the example.

\polylongdiv [stage=1] {(X-1)(X^2+2X+2)+1}{X-1}
\polylongdiv [stage=2] {(X-1)(X^2+2X+2)+1}{X-1}
\polylongdiv [stage=3] {(X-1)(X^2+2X+2)+1}{X-1}
\polylongdiv [stage=4] {(X-1)(X^2+2X+2)+1}{X-1}

Figure 2: Stepwise polynomial long division. The whole division is shown with
stage=10. Note that other printing styles, see table 7, might require one more
stage to put the remainder next to the result

\polylonggcd {(X-1)(X-1)(X^2+1)} {(X-1)(X+1)(X+1)}

Figure 3: Euclidean algorithm with polynomials; the last nonzero remainder is a
greatest common divisor

\polyfactorize {(X-1)(X-1)(X^2+1)}

\polyfactorize {2X^3+X^2-7X+3}

\polyfactorize {120X^5-274X^4+225X^3-85X^2+15X-1}



Figure 4: Factorizations of some polynomials

2 Commands

The tables 5 and 6 list the user commands defined by this package. Each
 stands either for a control sequence to store the internal representation of the
result, or it stands for a previously saved result to operate with. The polynomials
<a> and <b> are 'verbatim' polynomials as you would type them in math mode:
you may use +, -, *, \cdot, /, \frac, (, ), natural numbers, symbols like e, \pi,
\chi, \lambda, and variables; the power operator ^ with integer exponents can be
used on symbols, variables, and parenthesized expressions. Never use variables in
a nominator, denominator or divisor.

print long division a/b (maybe partially)	\polylongdiv \polylongdiv
print Euclidean algorithm for gcd(a, b)	\polylonggcd \polylonggcd
print factorization of the polynomial a	\polyfactorize \polyfactorize

Table 5: High-level user commands. The optional argument of the \polylongdiv
command changes settings for the particular division

	\polyadd \polyadd
	\polysub \polysub
	\polymul \polymul
\polyremainder ←a mod b	\polydiv \polydiv
	\polygcd \polygcd
print polynomial a	\polyprint \polyprint

Table 6: Low-level user commands

Note, however, that the support of symbols is very limited and that there is
neither support of functions like sin(x) or exp(x), nor of roots or exponents other
than integers, for example π or e^x. For teaching purposes this shouldn't be a
major drawback. Particularly because there is a simple workaround in some cases:
the package doesn't look at symbols closely, so define the function or 'composed
symbol' like as a symbol. For example, you could write

\newcommand\epowerx{e^x}
\[\polylongdiv{\epowerx x^3-\epowerx x^2+\epowerx x-\epowerx}{x-1}\]

Here the quotient and remainder are written next to dividend and divisor. This
'style=B-feature' is discussed in the next section.

Let's conclude this section with an example of the low-level commands. If we
want to divide by X-1 and print quotient and remainder,
we could do it like this:

\polydiv\polya {(X^2+X+1)(X-1)-\frac\pi2} {X-1}
'The quotient is \polyprint\polya,
the remainder \polyprint\polyremainder.'

Of course, we all know the result, so it isn't shown here. The calculation alone
could also be done by, for example,

\polymul\polya {X^2+X+1} {X-1}
\polysub\polya \polya {\frac{\pi}{2}}
\polydiv\polya \polya {X-1}

3 Keys

\polyset Predefined variables are X and x, the default style for printing long division is
shown in Figure 1 (b), and the left and right delimiters are \left( and \right).
The macro \polyset changes such default settings and accepts a comma separated
list of 'key=value' pairs. This is implemented using the keyval package [3]. Possible
keys and values are given in table 7. To make the style selection C clear and to use
variables and delimiters other than the default, look at the output of the following
example.

\begin{center}
\polyset{
vars=XYZ\xi, % make X, Y, Z, and \xi into variables
style=C, % use nonstandard style
delims={[}{]}}% nongrowing brackets
\polylongdiv{(\xi-1)(\xi^2+2\xi+2)+1}{\xi-1}
\end{center}

Afterwards previous settings are restored since the changes are made inside the
center environment or, more generally, inside a group. The same \polylongdiv
command now makes no sense. A constant would be divided by a constant.

Table 7: Keys and values. The delims key has in fact an optional argument for
use with growing delimiters. In this case you must specify invisible versions of the
two delimiters. The default is delims=[{ \left.}{ \right.}]{ \left(}{ \right)}