Re: Dividing Polynomials with remainders
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Re: Dividing Polynomials with remainders
> this is stupid question,
>
> what is a polynomial? We have not covered that yet :(
>
> / | Jeff | /
Not a stupid question at all! Very few textbooks contain a good (or even a
correct) definition of polynomial.
The set of polynomials is the set of all functions that can be built from
copies of the identity function f(x) = x and constant functions g(x) = c for
some number c (positive, negative, rational, whatever..) using the operations
of addition or multiplication only.
Examples of polynomials are 3x - 2, 5xx (or 5x^2 if you like), (xx+7x+77)^777,
etc. Strictly speaking, these are polynomial _expressions_, and there may
easily be two different expressions representing the same polynomial. Any
polynomial expression can be _expanded_ (using the laws of algebra) and
written in standard form. But it is often more convenient _not_ to expand a
polynomial expression.
I sometimes give my students a "polynomial construction kit" (a set of
function boxes and operation boxes and all the necessare "plumbing") and we
explore such questions as the most efficient way to build a particular polynomial.
Polynomials are "nice" functions to work with, and in practice people do
things like find polynomials that are _close_ to other functions (the topic is
approximating polynomials). Beginning courses in calculus often concentrate
on the calculus of polynomials only. Etc. etc.
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<