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Is $$\dfrac{x^2 + 2x}{x}$$ a polynomial?

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    What are your thoughts? – 2017-01-05
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    One argument against it being a polynomial would be that the domain doesn't include zero, and a polynomial has always domain = $\mathbb{R}$ – 2017-01-05
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    And so $ \frac{9}{3}$ is not an integer ? – 2017-01-05
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    @SeñorBilly A polynomial does not "always" have domain $\mathbb{R}$. Remember, domains are specified, not intrinsic. – 2017-01-05
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    There are some good reasons not to define a polynomial as a function. – 2017-01-05
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    So it seems the answer depends on your definition of polynomial. – 2017-01-05
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    @TonyK Definitely. If the expression is considered as an element of $\mathbb{R}(x)$ (the field of fractions of the polynomial ring $\mathbb{R}[x]$), then it is certainly a polynomial, because we can “cancel out the $x$”. As a function of a real variable, it is not a polynomial function, because its domain is not $\mathbb{R}$. – 2017-01-05
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    @user1952009 You're comparing apples and oranges. – 2017-01-05
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    It's definitely a polynomial almost everywhere :P – 2017-01-05
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    @egreg $\frac{9}{3} =3$ is really the same as $\frac{x^{2}}{x} = x$ – 2017-01-05
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    @user1952009 Depending on *where* you're working; in the framework of real functions of a real variable, they're not the same. – 2017-01-05
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    @user1952009 Not really: 9/3 = 3 is _always true_; $x^2/x=x$ is only correct if $x \ne 0$; since the left-hand side is undefined for $x=0$. – 2017-01-05

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Some people would say that the rational number $7/1$ is not really equal to the integer $7$, but merely canonically identified with it. But (after reaching a certain level of sophistication) mathematicians say that $7/1$ and $7$ are indeed equal.

Some people would say that the rational function $$ \frac{x^2+2x}{x} \tag{*}$$ is not really equal to the polynomial $$ x+2, \tag{**}$$ but merely canonically identified with it. But (after reaching a certain level of sophistication) mathematicians say that (*) and (**) are indeed equal.

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    It may depend on the context, but there is some subtlety to this analogy (as pointed out in the comments). If you, after canonical identification, indeed would say they are *equal*, then what about (what happens at) $x=0$...? – 2017-01-05
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Well, it's a polynomial in the variable $t=\tfrac{x^2+2x}{x}$... But you probably mean a polynomial in the (real?) variable $x$. What precise definition of polynomial are you using?

I would say no, because it is not of the form $$a_0+a_1x+a_2x^2+\ldots+a_nx^n$$ for any $n \in \mathbb{N}$ and real numbers $a_i$ ($0 \le i \le n$).

Note that you cannot just simplify $$\frac{x^2+2x}{x} = x+2$$ as this equality is only valid for non-zero $x$, so not for all $x$.