What actually is a polynomial? I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial.
I wasn't in the advanced mathematics class in 8th grade, then in 9th grade I skipped the class and joined the more advanced class. This question isn't about something I don't understand; it's something I missed.
My classes have not covered what really a polynomial is. I can generate one, but not define one. The internet has yielded incomplete definitions: "Consisting of multiple terms" or "A mathematical expression containing 2 or more terms and variables."
Take the following expressions for example:
$2x^2-x+12-2x^2+x-12$. Consists of multiple terms, but can also be expressed as $0$. Is zero a polynomial?
What about $x^{-1}$? $x^{-1}$ have -1 zeroes?">I've been told this one isn't a polynomial, but I don't understand why.
Is $x^2+x+1-x^{-1}-x^{-2}-x^{-3}$? a polynomial? It contains both positive and negative exponents?

tl;dr: What actually is the mathematical definition of a polynomial? Is $0$ a polynomial, and why isn't $x^{-1}$ a polynomial under this definition?

This isn't a definition suitable for pre-calculus, but I would say that a polynomial in a variable $x$ is anything whose $n^\text{th}$ derivative with respect to $x$ vanishes everywhere (i.e. is equal to zero everywhere), for some integer $n \geq 0$.
The nice thing about this definition is that it talks about how the polynomial behaves, rather than how you write it (so $\cos(2 \cos^{-1} x)$ is also a polynomial in $x$). It also generalizes appropriately to more abstract objects such as rings, functions, etc. as long as you define derivatives appropriately.
