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noelevans
2013-01-19 13:13:50

A polynomial in x is the result of a finite number of additions and/or multiplications of -1 and x.
Example:  3x² - 2x + 1 = (-1*-1 + -1*-1 + -1*-1)*x*x + (-1 + -1)*x + (-1*-1)
Example:  0 = -1*-1 + -1
Example:  -x^3 = -1*x*x*x

Resulting polynomials involve integral coefficients and non-negative integral exponents on x.

If one traces back each defined term until nothing but undefined terms are present, then terms
like "expression," "independent variable," "form," "variable," "denominator," and "function" would
create a succession of definitions terminating in a terribly convoluted definition of polynomial in
terms of just undefined terms (not unlike "well formed formulas" in logic).

Typically the best communication is obtained by tailoring the discussion to the intended audience.
Too much detail or not enough detail (especially in proofs) makes it difficult to follow.  And sometimes
when two (or more) people think they have finally arrived at a good understanding of what they have
been discussing, they later find out that they really had not grasped what the other was trying to get
across.   Each used perhaps the same words, but in the back of their mind had quite different ideas
as to what the words meant.  This can be especially troublesome when trying to "flesh out" a new
concept or new area of mathematics.

And therein lies much of the FUN in doing math.  Communicating with each other and trying to
figure out what in the world is going on!  Two minds (and the more the merrier) are "better than
one."  What one says usually sparks different thoughts in another's mind.  And back and forth
the exchange goes quite often culminating in some interesting stuff.  It's probably quite closely
akin to graffiti.

debjit625
2013-01-18 18:50:02

Thanks everybody...

anonimnystefy
2013-01-18 15:43:19

Hi noelevans

A polynomial is an expression which contains one or more independent variables and can be written in a form that doesn't have any variables in a denominator of a function.

noelevans
2013-01-18 15:32:46

Sometimes mathematicians play "fast and loose" with definitions.  We probably will not find total
agreement about what a "polynomial" is.  Some books will define "a polynomial in x" as opposed
to just a polynomial.  And what about a "polynomial in two variables?"  If y=2+x is x+y a binomial?

And what does "quotient" refer to?  If we consider 7÷2 as a fraction, is 7/2 a quotient?  Or is it
3.5?  Or is the quotient 3 (with remainder 1)?

And what is a fraction?  Which among the following are fractions?

x, 2/x,  x/y,  2/3, 2÷3, x÷y, 2*(1/x), x/1, x/3, xy where y=1/z, z where z=1/y, etc.

Let x=1/y,  y=1/3 and z=1/x.   Which are fractions?  x, y, 1/x, 1/y, y/3, xy, xz, yz, 1/yz, etc.

What is an arithmetic fraction, an algebraic fraction, etc.?

And in geometry is an equilateral triangle also isoseles?  There has been disagreement on whether
to make the set of equilateral triangles a subclass of isoseles triangles or a separate category of
triangles.

The best we find at times is a "local" definition where an author defines a term precisely for his
following discussion.  And often other mathematicians may find fault with this.

Mathematics is a LANGUAGE and has not been (nor ever is likely to be) "nailed down" so as to be
without ambiguity or disagreement even for the most common and "simple" concepts.

Sometimes we just have to "roll with the punches" and at times ask for clarification of what the
author intends (as is often the case in this forum).

Often pushing for the exact meaning of all the terms we encounter may result in returning to the
undefined terms of a system.  But then the expression we obtain for a "higher level" concept's
definition may be so long and involved with the elementary undefined terms as to be basically
incomprehensible.

As an example consider the "Sheffer stroke" or "Dagger" in T/F two valued logic.  Each of these
can be used to define the usual AND, OR, if..then, if and only if, NOT, exclusive OR.  But the
expressions for some of these are quite long and complicated using just the one stroke or dagger.
It is interesting that one "operation" can be used to define all the usual stuff, but it is way too
unwieldly to want to use it.  As humans we work better with the AND, OR, etc.

As another example consider binary vs hexadecimal.  Working with hexadecimal is much easier
for us humans than working with binary, especially when numbers are fairly large.  The strings
of 1's and 0's just get too long and difficult to deal with.

1/2(grateDAY )!

bobbym
2013-01-18 13:25:22

Hi;

Yes, x + 1 /x is not a polynomial.

anonimnystefy
2013-01-18 10:13:52

I think you're right...

debjit625
2013-01-18 04:07:17

In a book ,I saw this statement

"An algebraic expression containing two terms is called a binomial expression.
For example

,
, etc are binomial expressions.
Similarly, an algebraic expression containing three terms is called a trinomial."

Is it correct to say "an algebraic expression containing two terms is called a binomial expression or
three terms is called a trinomial" as binomial and trinomial are polynomials and a polynomial is made up of terms that are only added, subtracted or multiplied. and I think

is not a polynomial.