# What's the difference between these two definitions of polynomial function?

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?

• 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.
– Seth
Commented Jan 28, 2015 at 19:41
• I prefer the second and would like to define the degree of the polynomial at the same time; i.e. say something about $a_n$. Commented Jan 28, 2015 at 19:46
• 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). Commented Jan 28, 2015 at 19:48
• 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.
– Seth
Commented Jan 28, 2015 at 20:02
• The first definition makes no sense. It's like saying "Given integers $p,q$ with $q \ne 0$, a rational number is $p/q$." Commented Jan 29, 2015 at 15:03

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.