Past exam of the options I course of the University of Oxford / 2006 / P / P1

The Lagrange interpolation polynomial constructs existence explicitly:Distinct nodes make every denominator nonzero, and ${\ell }_j(x_k)={\delta }_{jk}$, so $p(x_k)=f_k$ and $p$ has degree at most $n$. If two such polynomials existed, their difference would have degree at most $n$ but at least $n+1$ distinct roots. The root bound for a nonzero polynomial forces that difference to vanish. Thus the interpolating polynomial exists and is unique.

The polynomial $p(x)=x^2-2x$ already has degree at most three and gives the specified data at every node. Uniqueness of polynomial interpolation therefore gives

Let $f(x)=x^4-6x^3+11x^2-5x$. The node polynomial isIt vanishes at all four nodes, so subtracting it leaves the data unchanged. Since $f(x)-\omega (x)=x$, uniqueness of polynomial interpolation gives