Polynomial used in Interpolation

Questions
1

Dear friends,

How could I get the polynomial used in the Interpolation command? What is, for example, the polynomial used in the following interpolation:

data = Table[{i, RandomReal[i]}, {i, 1, 10}];
f = Interpolation[data];

What was the order used in the “f = Interpolation[ data]”?
Thanks,
Ana

2

The default method is “Hermite”
See how-does-interpolation-really-work
If you want to force a polynomial to be used, you can use the Fit command. (but not a good idea to use polynominal to fit many points, else too much oscillation in between points, see Runge’s phenomenon) that is why Spline or Hermite would be better methods.

3 
data = Table[{i, RandomReal[i]}, {i, 1, 10}];
InterpolatingPolynomial[data, x] // Expand

Out[] = 1227.24 - 3354.85 x + 3658.58 x^2 - 2130.21 x^3 + 740.403 x^4 - 
        161.038 x^5 + 22.1034 x^6 - 1.8587 x^7 + 0.0873782 x^8 - 0.0017578 x^9
4

Thank you for all