Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 4 / 8D (Numerical Analysis)

The successive divided differences for nodes $-1,0,1,3$ areThus the Newton interpolation polynomial isExpanding the products gives the power formThe four values verify the data. Uniqueness follows because the difference of two degree-at-most-three interpolants would have four distinct roots.

Keep the original four-node interpolation polynomial $p$. The correction must vanish at all four original nodes, so a degree-at-most-four interpolant has the formSince $p(2)=-1$ and the product at $2$ is $-6$, matching the new value gives $-1-6c=-7$, hence $c=1$. Expanding yieldsIt takes the values $-7,-3,-3,-7,9$ at the five nodes. The same root-count argument proves uniqueness of this interpolation polynomial.