Re: how-to with TI86 please?


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Re: how-to with TI86 please?



Actually, a polynomial (in the variable x) is any expression that can be formed
by choosing copies of x, real numbers (any kind -- positive, negative,
integers, rationals or worse) or previously-built polynomials and applying the
operation of addition or multiplication.  This is a recursive definition -- x
is a polynomial, any real number is a polynomial, and the sum or product of any
two polynomials is a polynomial.  (More generally, we can allow complex numbers
instead of just limiting ourselves to real numbers).  Notice that the
_quotient_ of two polynomials is not necessarily a polynomial (though with luck
it may be).

Any such expression can be shown to be equivalent (by algebraic expansion --
this is the sort of thing that the TI-92 does well) to a polynomial expression
in _standard form_ -- a sum of terms each of which is a number times an integer
power of x.  Polynomials are easy to understand and work with, which is why
they have figured so largely in mathematics over the last 500 years or so.
Most of the simple hand-calculations that can be done with pencil and paper are
polynomial-based computations.

Calculators (especially symbolic manipulators like the TI-92) make it more
feasible to move beyond polynomials on an elementary level, but polynomials and
polynomial functions remain important.  Approximation of functions by
polynomials (Taylor expansions or Weierstrass approximations), for example, is
still a critical topic.  The fact is that polynomials are just plain easy to
work with!  This is one of the reasons why a general polynomial-solving routine
was included on the TI-85 and TI-86.

It should be noted that he polynomial equation being discussed in the last
couple of related messages here, X^4-999.4*X^3-599.89*X^2-109.994*X-6=0 ,
contains coefficients that seem to be _approximate_ numbers.  This means, of
course, that any roots that can be obtained must be approximate also.  Just
what precision can we state the roots to -- one place, two places. more?  This
is the sort of question that never used to get asked at all, because it was too
hard to get at any reasonable answer.  With modern calculators it is easier to,
say, perturb the coefficients slightly to explore such a question.  This is one
reason why we _should- have students using calculators in math class, so we can
get to all of those good hard questions!  :-)}


          RWW Taylor
          National Technical Institute for the Deaf
          Rochester Institute of Technology
          Rochester NY 14623

          >>>> The plural of mongoose begins with p. <<<<


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