The way I do this is to start with
$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\sim b_nx^n+b_{n-1}x^{n-1}+\cdots+b_1x+b_0=q(x).$
If I want these to be equal I want their outputs to be equal and in particular at $x=0$:
$p(0)\overset{!}{=}q(0)\Rightarrow a_0=b_0.$
Also I want all of their derivatives to be equal if they are to be equal. Direct calculation shows that
$\begin{align} p'(x)&=a_1+\mathcal{O}(x) \\ p''(x)&=2a_2+\mathcal{O}(x) \\ \vdots& \\ p^{(k)}(x)&=k!a_k+\mathcal{O}(x), \end{align}$ and similarly $q^{(k)}(x)=k!b_k+\mathcal{O}(x).$
Setting $p^{(k)}(0)\overset{!}{=}q^{(k)}(0)\Rightarrow k!a_k=k!b_k\Rightarrow a_k=b_k,$ for all $k$.