Is there a concrete definition/formula for finding the leading coefficient of any polynomial?

Question

Is there a concrete definition that tells one the leading coefficient of any polynomial? Using logic, I derived this formula:

$$ a=\frac{\frac{d^p}{dx^p}f(x)}{p!}$$

where $f(x)$ is a polynomial, $p$ is the power of the polynomial, and $a$ is the leading coefficient, but I have a feeling this isn't what I'm looking for, as I feel I'm missing some sort of major flaw. So, what is this concrete definition of the leading coefficient of any polynomial?

Also, I apologize if I'm using terminology incorrectly and making things confusing.

Answer 1

This one also works:

$$\lim_{x\to\infty}\frac{f(x)}{x^p}$$

Answer 2

Subtle point: a polynomial isn't the same than a polynomial function. In algebra, a polynomial is an expresssion $a_nx^n+\cdots+a_0$ with the $a_i$ in some field/ring/... If $a_n\ne 0$, then by definition $a_n$ is the leading coefficient. In a field of characteristic zero (like $\Bbb R$) different polynomials give different polynomial functions, but in positive characteristic, different polynomials can give the same polynomial function.

Your formula is true even in the algebraic case (see https://en.wikipedia.org/wiki/Formal_derivative), but in the algebraic case the coefficients are known beforehand.