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05AB1E, 10 bytes


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Takes x as first argument, n as second and p in ascending order as third.


s          # Forces the top two input arguments to get pushed and swaped on the stack
             (This is in case n is 0 so x is on the stack)
 F         # Do n times...
  ³         # Push the third input (the coefficients)
   g        # Get the length of that array...
    ݨ      # and create the range [0 ... length]
      m     # Take the result of the last iteration to these powers (it's just x for the first iteration)
       ³*   # Multiply the resuling array with the corresponding coefficients
         O  # Sum the contents of the array
           # Implicit print