Is This Function a Polynomial?

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Discussion Overview

The discussion revolves around determining whether the function $2x^{-3}-2x^{2}+3$ qualifies as a polynomial function. Participants explore the criteria for polynomial functions, including aspects such as degree, type, and leading coefficient, while also considering the implications of negative powers and asymptotes.

Discussion Character

  • Debate/contested, Conceptual clarification

Main Points Raised

  • One participant suggests the function is a polynomial, citing the sum of powers in the variable $x$ and identifying the highest power as 2, but expresses uncertainty about its type and mentions the presence of a vertical asymptote.
  • Another participant argues that polynomials cannot have negative powers of $x$, asserting that the function is not a polynomial and highlighting properties of polynomials such as continuity and differentiability.
  • A third participant reinforces the criterion that exponents must be non-negative integers for an expression to be classified as a polynomial, prompting others to reconsider their stance based on this information.
  • A later post mentions an unrelated project involving precalculus problems, which does not directly contribute to the discussion about the polynomial classification.

Areas of Agreement / Disagreement

Participants disagree on whether the function qualifies as a polynomial, with some asserting it is while others firmly state it is not due to the presence of a negative exponent.

Contextual Notes

There is a lack of consensus on the definition of polynomial functions, particularly regarding the treatment of negative exponents and the implications for continuity and differentiability.

karush
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$2x^{-3}-2x^{2}+3$

Decide whether the function is a polynomial function, If so, states its degree, type, and leading coefficient.

well, I presume it is a polynomial since it the sum of powers in the variable x. Its degree is 2 since that is highest power, but I don't know its type, since it is not quadratic or cubic. and has a vertical asymptote. and of course the leading coefficient is 2

couldn't find the answer in the book?
 
Last edited:
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Polynomials cannot have negative powers of $x$, so it's not a polynomial. Polynomials are continuous everywhere, differentiable everywhere, are well-behaved and have no asymptotes.

See the Wikipedia article, it says "non-negative integer exponents" :)
 
One of the criteria for an expression to be a polynomial is that the exponents must be non-negative integers. So given that piece of information, what do you say the answer is?

Here's a site that goes over some examples.
 
this is a overleaf project I am working lots of pre calc problems
140 pdf 100 pages
 

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