The following introduces a method of immediately deriving a polynomial that passes through an arbitrary number of points. The key to this solution is that we want an exact fit at the points given and we don't care what happens inbetween those points . The general solution is To see how this works, consider the product term.
All bets are off regarding the behaviour between the fixed points.
The polynomial is of degree N and could violently fly off anywhere. The continuous curve for the numerical example above is shown below. I am providing some hints. The following is the solution to the posted problem, although it probably doesn't offer the insight you are seeking, it certainly falls within the scope of competition. To illustrate my solution I will use the following symbols instead of the numbers you used simply to save space.