🚀 go-pugleaf

Murphy Waggoner writes

> The cubic formula is not as straight forwared as the quadratic formula
> and the quartic formula uses the cubic formula.  They do not take up
> one or more chalkboards to write down and could easily (for a relatively
> good programmer) be incorporated into a program.  This program will
> give the EXACT values for many polynomials (however, there are still
> some programming consideration to be made to achieve exact answers for
> MORE polynomials and it would be impossible to get exact answers for
> ALL polynomials).

There is a philosophical question lurking here, which was perhaps not
considered by the individual first suggesting calculator implementation
of the algorithms (discovered 400 years ago) for solving cubic and
quartic polynomial equations.  Even for very simple non-linear equations
such as xx = 2 it is normally impossible to derive an _exact_ root.
In this case, the roots are sqrt(2) and its opposite.  But what are
these numbers?? All we have is an _expression_ for the roots, and we
must resort to one or another algorithm to pursue calculation of the
numeric value of the root to a desired precision (try working out the
square root of 2 to a hundred places sometime!).

In the case of the expressions derived using the formulas for cubics
and quartics, the results are a fearsome thicket of radical signs,
with all sorts of special cases for real and complex roots -- in short,
hardly a _practical_ method of solving polynomial equations, even when
we have easy access (at the touch of a key) to computation of square
and cube roots.  These methods were of some practical interest 400
years ago, however, because they _did_ allow pursuit of roots that had
not been accessible before, especially when the roots happen to be
rational.  But this is a vanishingly small percent of the cases!

There was also the glimmer of possibility of a general solution in
radicals of _all_ polynomial equations.  However, that hope was
quashed when Abel proved (150 years ago) that no general solution
can exist for equations of degree 5 and higher.  The quadratic
formula remains extremely useful in both practical and theoretical
calculations, but it is very, very rare that one ever has need to
call on the cubic or quartic formulas.  As was pointed out, general
polynomial equation solvers (employing numerical algorithms) are a much more
practical approach when a root is needed.  One needs to be aware pecial
cases (like double roots) that "throw" these solvers exist, of
course, and be ready to deal with the difficulty  -- that's life in
the mathematical fast lane!  We are in a much better position to investigate
anomalies now that we can examine the _graphical_
behavior of functions, again at the touch of a key.

A final comment on the matter of solving polynomial equations is
needed -- that is the fact that your roots are only as good as your
coefficients!  If you are solving a theoretical equation where the
coeeficients are, say, exact integers, well and good -- go ahead
and pursue all the precision you wish.  But if your coefficients are
understood to be _approximate_ numbers (as is the case in most
practical applications) then there is no point wasting time on
theoretical solutions!!  By the way, this is one of the things that
ought to be changing as a result of general availability of
calculators such as the TI-82 and TI-85.  There is absolutely no
reason for pre-calculus mathematics (or calculus, either!) to concern
itself exclusively with polynomials with small integer coefficients,
is there?


        ||      RWW Taylor                 rwtnts@rit.edu
        ||      National Technical Institute for the Deaf
        ||      Rochester Institute of Technology
        ||      Rochester NY 14623