Basically it says that $$|x|>1,2n|a_{n-1}|,\ldots,2n|a_0|$$
Does this mean that $$|x|>\max(1,2n|a_{n-1}|,\ldots,2n|a_0|)$$
It seems to be essential when proving that polynomial whose degree (namely $n$) is odd has a root.
Reference: Spivak Calculus,chapter 7 page 111
You guessed correctly, it means that $|x|>1\land x>2n|a_{n-1}| \land \ldots \land x>2n|a_0|.$
Or equivalently, as you suggested, $|x|>\max(1,2n|a_{n-1}|,\ldots,2n|a_0|)$.