Exercise 13, § 2

Let $a, (1 \leq i \leq n)$ be $n$ distinct elements of a commutative field $K$, $\beta_i$ ($1 \leq i \leq n$) and $\gamma_i$ ($1 \leq i \leq n$) 2n arbitrary elements of $K$. Show that there exists one and only one polynomial $f \in K[X]$ of degree $\leq 2n - 1$ such that $f(\alpha_i) = \beta_i$ and $f''(\alpha_i) = \gamma_i$ for $1 \leq i \leq n$ (Hermite interpolation formula) (to begin with consider the case where $2n - 1$ of the $2n$ elements $\beta_i, \gamma_i$ are zero).