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.

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.

]]>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 )!

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

]]>"An algebraic expression containing two terms is called a binomial expression.

, , etc are binomial expressions.

For example

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