Let be $n$ points in the plane ($x_i\neq x_j$ for $i\neq j$ ). Then there exists a unique polynomial $p(x)$ of degree at most $n-1$ such that $y_i=p(x_i)$ for $i=1,\ldots,n$ .

Such polynomial can be found using Lagrange's interpolation formula: $$ p(x)=\frac{f(x)}{(x-x_1)f'(x_1)}y_1+\frac{f(x)}{(x-x_2)f'(x_2)}y_2+\cdots+\frac{f(x)}{(x-x_n)f'(x_n)}y_n $$ where $f(x)=(x-x_1)(x-x_2)\cdots(x-x_n)$ .

To see this, notice that the above formula is the same as