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#1 2017-07-31 18:45:56

Abbas0000
Member
Registered: 2017-03-18
Posts: 11

Polynomial

Why something with negative exponent or a variable at denominator is not allowed to be called polynomial?(I know the rules of polynomial but I don't know why )

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#2 2017-07-31 20:47:30

zetafunc
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Registered: 2014-05-21
Posts: 2,107
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Re: Polynomial

If negative exponents were allowed, then polynomials would also have poles. The fact that we take polynomials to only include non-negative exponents allows their domains to include all of
and means that they are closed under addition, multiplication, and even differentiation and integration. Note that the latter isn't the case if we allow negative exponents: it isn't necessarily the case that the integral of a rational function (say,
) is also contained in your list of polynomials (it isn't, for this example). However, you might like to look at something like the ring of rational functions, where your negative exponents are permitted (and satisfy many nice structural properties). Laurent series in complex analysis also include negative powers.

Morally, letting negative powers be included in the definition of a polynomial introduces the notion of poles: we like polynomials because they are some of the simplest, most well-behaved objects to study which are differentiable everywhere. Not the case with negative exponents!

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#3 2017-08-02 16:49:33

Abbas0000
Member
Registered: 2017-03-18
Posts: 11

Re: Polynomial

So ,,, we don't include negative exponents or a variable in denominator a part of polynomial just because we do not want to make polynomials hard to understand or a difficult thing . Is it just right for square root of a variable?

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#4 2017-08-02 20:14:57

zetafunc
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Registered: 2014-05-21
Posts: 2,107
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Re: Polynomial

It's not necessarily a case of not wanting them to be hard to understand -- as I pointed out, the moment you want to include negative exponents, you lose the nice properties that polynomials with non-negative powers have.

Abbas0000 wrote:

Is it just right for square root of a variable?

Not sure what you mean here -- we get similar issues if we start introducing square roots into polynomials too (and, thus, square roots of variables are not considered to be polynomials). For instance,
isn't differentiable at zero, nor is it even defined for negative numbers. Polynomials do not have this issue -- and that's only considering real variables. Thinking about complex variables, one then needs to clarify what is meant by "square root"!

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#5 2017-08-05 17:21:16

Abbas0000
Member
Registered: 2017-03-18
Posts: 11

Re: Polynomial

Thanks a lot !!:-)

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#6 2017-08-05 18:18:53

iamaditya
Member
From: Planet Mars
Registered: 2016-11-15
Posts: 725

Re: Polynomial

Hi Abbas0000,

A polynomial is an algaebric expression in the form of

which satisfies 2 conditions-

•The powers of the variables are whole nos. (In this case

);and
•The coefficients of the variables are Real nos. (In this case
)

So any expression of the type satisfying the 2 conditions are classified as a polynomial.

If the variables are non-whole nos. and the coefficients are non-real nos. then the expression becomes too complex to be classified as a polynomial.

Last edited by iamaditya (2017-08-05 18:22:41)


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#7 2017-08-05 20:13:41

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,107
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Re: Polynomial

That's not strictly true -- the coefficients can belong to any field, be it
or
. What you have described is only a polynomial belonging to
. More precisely, to say that the powers are whole numbers is not sufficient -- they need to be non-negative whole numbers.

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