# [05AB1E], <s>23</s> <s>22</s> 19 bytes

 ÄOsLŽ*{.Δ∞<m¹¤**Od

[Try it online!][TIO-lebeyhca]

Takes as an input the coefficients list, from the lowest power to the biggest one, and a number \$n\$, and returns \$a_n\$. 

If this is invalid because we may not assume \$\sin\$ has infinite precision, then the following 20 bytes alternative could be used, which uses a spacing of \$\frac1 n\$ for the \$x\$s:

 ÄO*(D(Ÿ²/.Δ∞<m¹¤**Od

## Explanation

Uses the fact that \$ \sum_{i=0}^m{|c_i|}\$ is a bound on polynomial roots (when \$|c_m| \geq 1\$, which is always the case for integer coefficients) to find the non-negative value of \$p(x) \cdot c_m\$ with the smallest \$x\$ among the values \$ x = bound \cdot \sin(i) \$ for \$ 1 \leq i \leq n \$. This is correct since the sine of integers is dense in the range \$[-1, 1]\$, so we can get arbitrarily close to the target number, and since we always keep \$x\$s as \$n\$ increases the sequence is non-increasing.

```
Ä Absolute value of each coefficient
O Sum the absolute values
s Swap with, so now n is on top
L Get the range [1, 2, 3, ..., n]
Ž Calculate the sine of each number in that range
* Multiply those numbers by the bound
{ And sort them
.Δ Now find the first value such that
 ∞< Push the infinite list [0, 1, 2, 3, ...]
 m Calculate x^n for each n in the list
 ¹ Get the polynomial's coefficients
 ¤ Get the last coefficient, without popping
 * Multiply it with all of the coefficients
 * And then multiply that with x^0, x^1, ...
 O Calculate the sum of those products
 d And check if it is non-negative
```

[05AB1E]: https://github.com/Adriandmen/05AB1E
[TIO-lebeyhca]: https://tio.run/##yy9OTMpM/f//cIt/sc/h1kN7tar1zk151DHPJvfQzkNLtLT8U/7/jzbU0TXSMQBCXSDLGEgbxXIZGgABAA "05AB1E – Try It Online"