Contents - Student subdomain for University of Bath

1.2. EXPANSION AND SIMPLIFICATION 19
1.2.2 Expansion
This word is relatively easy to define, as application of the distributive law, as
seen in the first bullet point of Maple’s description.
• The expand command distributes products over sums. This is
done for all polynomials. For quotients of polynomials, only
sums in the numerator are expanded; products and powers are
left alone.
• The expand command also expands most mathematical functions,
including . . ..
In any given system, the precise meaning of expansion depends on the underlying
polynomial representation used (recursive/distributed — see page 35), so Maple,
which is essentially distributed, would expand x(y+1) into xy+x, while Reduce,
which is recursive, would not, but would expand y(x + 1) into xy + y, since its
default ordering is ‘x before y’.
Expansion can, of course, cause exponential blow-up in the size of the expression:
consider (a + b)(c + d)(e + f) . . ., or sin(a + b + c + . . .). The second
bullet point of Maple’s description can lead to even more impressive expansion,
as in
expand(BesselJ(4,t)^3);
(just where did the number 165888 come from?) or
expand(WeierstrassP(x+y+z,2,3));
1.2.3 Simplification
This word is much used in algebra, particularly at the school level, and has
been taken over by computer algebra, which has thereby committed the sin of
importing into a precise subject a word without a precise meaning.
Looking at the standard textbooks on Computer Algebra Systems
(CAS) leaves one even more perplexed: it is not even possible to find
a proper definition of the problem of simplification [Car04].
Let us first consider a few examples.
1. Does x2 −1
x−1
simplify to x + 1? For most people, the answer would be
‘yes’, but some would query “what happens when x = 1”, i.e. would
ask whether we are dealing with abstract formulae, or representations of
functions. This is discussed further for rational functions on pages 42 and
198, and in item 5 below.