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Apologies if this question is too trivial. I am having trouble precisely defining polynomials. All of the definitions I have seen say that expressions of the form $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ (standard form) are polynomials. But what about unsimplified expressions that are not of the standard form, but are equal to (can be rewritten as) an expression in standard form? Additionally, are expressions that look different but are equal to each other considered the same or different expressions?

Examples:

  1. $\frac{x^2+4x^2+0}{x−x+1}=5x^2$. RHS is clearly a polynomial, but the LHS?
  2. $2x^5+x^{10}-x^{10}+x^0-1x^0=2x^5$. RHS is a monomial. Is LHS a monomial or a sum of monomials?

A similar question is this, but none of the answers really address whether non-standard-form expressions are indeed polynomials.

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    Personally, when trying to figure out whether something is a polynomial, I simplify it first, before going from there. However, the example LHS is not the exact same as the example RHS, since the example LHS is undefined for $x=0$. For values $x \neq 0$ LHS=RHS and they are both polynomials. If an expression looks different to a polynomial but is the same for all values, then the expression in question will be a polynomial. – GSmith Jul 17 '24 at 08:47
  • @GSmith Thanks for pointing out the error, I have replaced it with an equality that is true for all reals. –  Jul 17 '24 at 08:50
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    Is $1+1$ a natural number? – ancient mathematician Jul 17 '24 at 08:52
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    @user985091 From my understanding, if you can simplify an equation seemingly not an polynomial, to make it one without affecting its domain, then the equation will be a polynomial. – GSmith Jul 17 '24 at 08:53
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    For the second example, I'd say the LHS is just a monomial, as from my (not necessarily correct) understanding, most properties are typically considered after simplifying. – GSmith Jul 17 '24 at 09:01
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    Books that are a bit more careful (school books often are not, but many college algebra books are) will usually say something along the lines of what @GSmith said. In math it is fairly standard to include the phrase "can be expressed as" in definitions when being more careful, such as a rational number is a real number that can be expressed as a quotient of integers (even if it is not presented as such). Of course, in the case of expressing functions, the "can be expressed" part includes domains, so something whose domain is NOT all real numbers will not be a polynomial. – Dave L. Renfro Jul 17 '24 at 09:04
  • I just looked through several college algebra books on my bookshelves and none used a phrase along the lines of "that can be expressed". I suppose I'm remembering this because that's what I always said when I used to teach. In the books I looked at, I did see several that defined what a "polynomial function" is, saying it is a function that is $[\cdots]$ Such a definition falls back on the meaning of a function, and of course such books are always careful to distinguish between the function itself and how it is represented (table form -- if domain is a finite set, formula, etc.). – Dave L. Renfro Jul 17 '24 at 09:17
  • An example for what is NOT a polynomial is $\frac{x^2}{x}$ because this is only $x$ for $x\ne 0$. – Peter Jul 17 '24 at 09:17
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    This is a very difficult question to answer because polynomials arise at every level of mathematics, with several competing definitions. You are right that the idea of a polynomial simply being an "expression" is rather imprecise, but to give a precise definition of a polynomial likely requires more mathematical machinery than you are familiar with. – Joe Jul 17 '24 at 09:23
  • @DaveL.Renfro Yeah, the whole "can be expressed as" phrase seems to have disappeared from any of the calculus and precalculus texts I have taught out of in the last decade. I'm generally very careful to put those phrases in my course notes, but the books have kind of lost the plot. – Xander Henderson Jul 17 '24 at 13:07

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