1. Using the Newton form, find the interpolating polynomial passing through the points , , , and .
To find the four coefficients, we can use the divided differences.
Thus, the four coefficients are and the upper diagonal of the calculated divided differences.
Using the Newton Forward Divided Difference formula, we find the polynomial is
- .
2. Find the interpolating polynomial passing through the four points given in the exercise above as well as through the point
To find the new coefficient, we can simply add the new point to the end and calculate .
Thus, we find the polynomial is
- .