I'll give a couple of examples. This picture here
shows how to rationalize a fraction if there is a sum of three radicals on the bottom.
The next picture linked is a look at Horner's
process where division of a linear polynomial
is used to find remainders from a degree n
polynomial, which are coefficients of a
transformed polynomial...the same work can be
done by using derivatives just above the method.
It may have been possible that given time and
some play anyone could have found it out.
Maybe someone is lucky to see these done in
class. But as far as I know and experienced,
this stuff isn't talked so much. Instead, we got
rationalizing sum/difference of two radicals and