Computing a specific coefficient in a quotient of polynomials
MATL, 6 bytes

Y-PiQ)


The inputs contain the coefficients in order of decreasing powers.

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Explanation


  (De)convolution is the key to success


     % Implicit inputs: two numerical vectors
Y-   % Deconvolution 
P    % Flip
i    % Input: integer
Q    % Add 1
)    % Get the entry at that position
     % Implicit display