Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials

A formal real polynomial is either the zero polynomial $0$, or a finite coefficient list $(a_0,\dots ,a_n)$ with $a_n\mathrm{\ne }0$; we write the latter as $p(X)={\sum }_{k=0}^n{a}_k{X}^k$. The list, rather than the function it induces, is the polynomial object. Its evaluation at $x\in \mathbb{R}$ is the real number $p(x):={\sum }_{k=0}^n{a}_k{x}^k$.

For nonzero $p=(a_0,\dots ,a_n)$, define $\mathrm{deg}p:=n$ and $\mathrm{lc}(p):=a_n$. The zero polynomial has no degree and no leading coefficient. A nonzero polynomial is monic when $\mathrm{lc}(p)=1$. Thus degree and leading coefficient are defined from the displayed finite list, without asserting that distinct formal polynomials define distinct functions. The conventions for finite sums and integer powers used in the evaluation are recorded in Finite sums and finite products, by recursion, Integer powers $a^m$.