 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2013/p/1/a/solution

p (x) = j = 0 \sum n f (x_{j}) ℓ_{j} (x), ℓ_{j} (x) = 0 \leq k \leq n k \neq = j \prod \frac{x - x _{k}}{x _{j} - x _{k}} .

The distinct nodes make all denominators nonzero. Each Lagrange basis polynomial has degree

n

and satisfies

ℓ_{j} (x_{i}) = δ_{ij}

, so the sum has degree at most

n

and interpolates every specified value.

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