Hermite cubic spline! No 9th degree polynomials allowed! Much more work. Bounding is harder....
Using linear algebra -- ideally you will think in terms of the matrices, rather than the equations that Mathematica wants to solve simultaneously. But I didn't take off for that.
Muller: Interesting that some got complex roots (I didn't). Muller can produce complex roots. In that case, you would use the modulus function to find the distance between (or size) of things.