> I just figured out a way to get a TI82 to do symbolic manipulation! The
> method will take any algebraic expression with degree 4 or less and
> simplify it to it's polynomial form.
>
> Here's what you do:
> type the expression to be simplified into Y1,
> use the table to generate a list of data points (at least one more than
> the degree of the answer),
> type the data points into the statistical table,
> run the appropriate regression (quartic for expressions of degree 4,
> cubic for degree 3 and quadratic for degree 2).
>
> I haven't decided what this is good for yet. At the least, I can use
> it for myself to generate expressions in polynomial form for homework
> or test questions. I'm going to see if one of my students can write a
> program for this. If anyone has a program based on this idea, please
> post it.
That's a neat trick. A way to get at this a bit more efficiently might
be to _evaluate_ Y1 at a set of distinct values, say 1, 2, 3, 4 and 5.
If Y1 is really a representation of a polynomial of degree 4 or less,
then Y1(1) will be the sum of the coefficients of the polynomial, say
a+b+c+d+e. Similarly, Y1(2) will be 16a+8b+4c+2d+e, and so forth.
Multiplying the 5x1 matrix containing the 5 evaluated results by the
inverse of the 5x5 matrix whose rows are [t^4 t^3 t^2 t^1 1] evaluated
at the same set of five values -- in the example we started, the first
row would be [1 1 1 1 1], the second row [16 8 4 2 1], and so on --
will give you the putative coefficients directly, and you can check by
evaluating Y1 and the polynomial you have derived at some other value
(pi, maybe?) All this would be very easy to program.
Of course that's just a naive cut, and likely would break down under
special circumstances (double roots, maybe?). This whole idea is closely
related to interpolation theory, something that never used to get into
the curriculum at all unless you were pursuing some specialty such as
numeric analysis. There's probably a lot of fun stuff lying out there
in the old texts that can now be done by students in introductory algebra
courses.
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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