 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ib/paper-4/8d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The successive divided differences for nodes

- 1, 0, 1, 3

are

x - 1 013 f [x] - 7 - 3 - 3 9 f [x, x_{next}] 406 f [x, x_{next}, x_{next next}] - 2 2 f [- 1, 0, 1, 3] 1

p (x) = - 7 + 4 (x + 1) - 2 (x + 1) x + (x + 1) x (x - 1) .

Expanding the products gives the power form

p (x) = x^{3} - 2 x^{2} + x - 3.

The four values verify the data. Uniqueness follows because the difference of two degree-at-most-three interpolants would have four distinct roots.

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