05AB1E, 23 22 bytes
ÄO*D(Ÿ²/Dε∞<m¹*O}düÊÏн Takes as an input the coefficients list, from the lowest power to the biggest one, and a number \$n\$, and returns \$a_n\$.
Uses the fact that \$ \sum_{i=0}^m{|c_i|}\$ is a bound on polynomial roots to generate the range \$ [-\sum_{i=0}^m{|c_i|}, \sum_{i=0}^m{|c_i|}]\$ with jumps of \$\frac1 n\$, and finds the last position the polynomial changes sign.
Ä Absolute value of each coefficient O Sum the squares * Multiply it by n D Duplicate that value ( And negate it Ÿ And then push the range of values between them ² Push n / And divide each number in that range by it. We now have the range [bound, -bound], with jumps of 1/n D Duplicate the range ε For each x in the range: ∞< Push the infinite list [0, 1, 2, 3, ...] m Calculate x^n for each n in the list ¹* Multiply it by the polynomial's coefficients O And calculate the sum of those products } d Is positive üÊ For every pair of items, check if their "is positive" is not equal Ï Keep the elements from the range's copy in indices where that is true н And take the first such value (and implicitly output it)