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For instance, why are only non-negative integer powers of a variable allowed within the scope of the definition of a polynomial? I am thinking there must be some reason within the developmental context of the definition but I am not sure what it is.

X Droid
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  • There is a differential barrier between the negative and positive powers of variables in a polynomial. Beyond that, what would the use be of a negative power in an integer-valued polynomial? For other definitions, you may consider the "binomial form" of a polynomial in place of the traditional exponent basis. – abiessu Oct 11 '19 at 04:02
  • Hi abiessu, what is a differential barrier? – X Droid Oct 11 '19 at 04:03
  • Take the derivative of a positive-integer-exponent polynomial. Can any exponent ever be negative? Take an integral over a negative-integer-exponent variable. Can the exponent become positive? This is a definition of a differential barrier. – abiessu Oct 11 '19 at 04:06
  • Okay, I am still a pre calculus student but I looked it up and I kinda understand this. But I still don’t see what that has to do with defining a polynomial the way that we do it. Also, I think that by ‘binomial form’ you mean the form of a polynomial which is similar to a binomial expansion, without the second term in each term (pardon my crude way of explaining but I am new here and don’t yet know how to use mathjax). The point I am trying to make is I understand the definition of a polynomial. I just don’t understand why a polynomial is defined that way. – X Droid Oct 11 '19 at 04:25
  • Can you explain that to me without using calculus because although I did play with calculus a little bit I am not proficient with it? – X Droid Oct 11 '19 at 04:26

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We want the polynomials to be the smallest set which contains all the numbers and a new symbol $x$, where we are allowed to add and multiply. We are not able to obtain negative or fractional powers of $x$, or any other construction by just adding and multiplying. However, we do get the polynomials this way.

vujazzman
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    Yes, this is it in a nutshell. – Lubin Oct 11 '19 at 04:49
  • Hi vujazzman. Query: your answer implies that there is a larger set of functions which allow for more variables and more complicated operations AND that polynomials is a subset of this larger set. Am I correct in making this assumption? If yes, then please tell me what this larger set is called. If not, then please advise me as to where I went wrong. – X Droid Oct 11 '19 at 05:26
  • In general a function is any assignment of values from one set to values in another. There are as many important classes of functions as there are people to imagine them. Just some examples of important sets of functions, say from the real numbers to the real numbers are analytic functions, differentiable functions, continuous functions, linear functions, and so on and so on. – vujazzman Oct 11 '19 at 05:36