Re: How do you define a polynomial? What is the official definition of a polynomial?
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Wed, 7 Dec 2005 20:49:28 +0000 (UTC)
In article <1133987693.198529.106690@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
comtech <comtech.usa@xxxxxxxxx> wrote:
>
>David C. Ullrich wrote:
>
>> >
>> >>What is the essential difference between a polynomial and some function
>> >>such as exp(x)?
>> >
>> >There are many differences in behavior. What do you mean by
>> >"essential"?
>>
>> Since I doubt that he'll say, I'll conjecture that all he meant
>> by the question was "exactly why is exp not a polynomial? What
>> property of exp allows me to verify this?" or some such.
>>
>
>Yeah, you said it... why exp(x) is not a polynomial... ?
Because exp(x) is a function, and polynomials are not functions.
One can use polynomials to define functions; these functions are
called "polynomial functions". Every polynomial corresponds to a
polynomial function, and in some settings, different polynomials
correspond to different polynomial functions (though not always).
You could be trying to ask "why is exp(x), as a function from the
reals to the reals, not a polynomial function?"
Because there is no polynomial a_0 + a_1*x + ... + a_nx^n such that
exp(k) = a_0 + a_1*k + ... + a_n k^n for every real number k.
You can see this any number of ways. It is easy to show that if f(x)
is a polynomial function, then its derivative exists and is also a
polynomial function; and that there exists a positive integer n such
that the n-th derivative of f(x) is zero. But the derivative of exp(x)
is exp(x), so there is NO positive integer n such that the n-th
derivative of exp(x) is zero. Therefore, exp(x) cannot be a polynomial
function.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx
.
- References:
- How do you define a polynomial? What is the official definition of a polynomial?
- From: comtech
- Re: How do you define a polynomial? What is the official definition of a polynomial?
- From: David C . Ullrich
- Re: How do you define a polynomial? What is the official definition of a polynomial?
- From: comtech
- How do you define a polynomial? What is the official definition of a polynomial?
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