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Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$

Definition 2: The function $p:\mathbb{R} \rightarrow\mathbb{R}$ is a polynomial function if there exist $a_n,...,a_1,a_0 \in \mathbb{R}$ such that $p(x)=a_nx^n+...+a_1x+a_0$ for any $x$

The first definition is the one I've always used; the second comes from Linear algebra done right. Now, I can understand both of them, but I can't see why one would need to complicate matters with the second definition, which seems a little more difficult to me. What's the difference? Why use one or the other?

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    $\begingroup$ The first definition is confusing and strange, it seems to imply that a "polynomial" with different coefficients than the chosen coefficients is not a polynomial. The second is the standard definition. $\endgroup$
    – Seth
    Commented Jan 28, 2015 at 19:41
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    $\begingroup$ I prefer the second and would like to define the degree of the polynomial at the same time; i.e. say something about $a_n$. $\endgroup$ Commented Jan 28, 2015 at 19:46
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    $\begingroup$ Definition 2 is a more precise way of stating Definition 1. If you think these definitions say different things, then you are misreading one of them (most likely, Definition 1, as it is sloppier). $\endgroup$ Commented Jan 28, 2015 at 19:48
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    $\begingroup$ Nicol, neither definition define the degree of a polynomial, because $a_n$ is not assumed nonzero. Definition 1 is supposed to say what definition 2 says, but either fails to be clear, or correct, depending on your interpretation. Just forget definition 1, and use definition 2. $\endgroup$
    – Seth
    Commented Jan 28, 2015 at 20:02
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    $\begingroup$ The first definition makes no sense. It's like saying "Given integers $p,q$ with $q \ne 0$, a rational number is $p/q$." $\endgroup$
    – TonyK
    Commented Jan 29, 2015 at 15:03

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I still don't really know what you are trying to say with definition 1, but I suspect you are trying to ensure some sort of uniqueness of coefficients, and apparently define the degree of a polynomial. I think you missed the mark here, so here is a simple and correct way to define a polynomial, and it's degree.

Define a polynomial of degree $n\geq0$ to be a function $p:\mathbb{R}\to \mathbb{R}$ such that there exists $a_n,\dots,a_0\in\mathbb{R}$ with $a_n\neq0$ and $p(x)=a_n x^n+\dots a_1 x+ a_0$ for all $x\in\mathbb{R}$. Define a polynomial of degree $n=-\infty$ to be the zero function.

Define a polynomial to be any function which is a polynomial of degree $n$ for some $n$.

The coefficients are unique in the following sense: if any function $p:\mathbb{R}\to\mathbb{R}$ satisfies $p(x)=a_n x^n+\dots a_1 x+ a_0$ for all $x$ and also $p(x)=b_m x^m+\dots b_1 x+ b_0$ for all $x$, then without loss of generality we may suppose $n\leq m$, and defining $a_i=0$ for $i=n+1,\dots, m$ we may conclude that for $j=0,\dots,m$, $a_j=b_j$. The uniqueness statement may be proven and is not part of the definition.

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    $\begingroup$ After some reflection I think I understand now. Thank you for your patience. $\endgroup$
    – Adrian
    Commented Jan 29, 2015 at 14:56

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