Name that polynomial … revisited

This game is thanks to David Bedford at the BCME Conference in Warwick.

One person writes down a polynomial with positive integer coefficients. Call it f(x) They then choose an integer that it bigger than all their coefficients. Call it n. They then calculate f(n) and give just n and f(n) to the second person.

The second person should be able to name that polynomial! How?

Example:

Given only n=8 and f(n) = 6855 how could you work out that the polynomial was

?

It might help to think about an example when n=10 first.

Here is a cheatsheet for when you have a strategy. Can you write some Python code to work out the polynomial for you?

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