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#1 2013-01-17 05:07:17

debjit625
Member
Registered: 2012-07-23
Posts: 89

Binomial expression

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.

Last edited by debjit625 (2013-01-17 05:09:49)


Debjit Roy
___________________________________________________
The essence of mathematics lies in its freedom - Georg Cantor

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#2 2013-01-17 11:13:52

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,861

Re: Binomial expression

I think you're right...


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2013-01-17 14:25:22

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,933

Re: Binomial expression

Hi;

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


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#4 2013-01-17 16:32:46

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Binomial expression

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 )! smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#5 2013-01-17 16:43:19

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,861

Re: Binomial expression

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.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2013-01-17 19:50:02

debjit625
Member
Registered: 2012-07-23
Posts: 89

Re: Binomial expression

Thanks everybody...


Debjit Roy
___________________________________________________
The essence of mathematics lies in its freedom - Georg Cantor

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#7 2013-01-18 14:13:50

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Binomial expression

How about something like:

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.  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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