obtain polynomials $P(y)$ for specific data

Question

assume $P(y)$ is a polynomial for following data :

$\big(y_i,y_i^2 (2y_i-1)\big)\,$ and $\,y_i=i ,\,i=0,1,2,3$

Then What is $P(y)$ equal to ?

Answer 1

Two polynomials $\,P(y)\,$ and $\,Q(y)\,$ of degree at most $\,n\,$ which are equal for $\,n+1\,$ distinct values of $\,y\,$ must be identical. This follows directly from the FTA because $\,P(y)-Q(y)\,$ is a polynomial of degree at most $\,n\,$, and therefore if it has $\,n+1\,$ roots then it must be the zero polynomial.

For $\,n=3\,$ and $\,Q(y) = 2y^3-y^2\,$, if $\,P(y)\,$ has degree at most $\,3\,$ then $\,P(y)=Q(y)\,$. If $\,P(y)\,$ has degree greater than $\,3\,$, it must differ from $\,Q(y)\,$ by a polynomial which is zero at the given points, so in the end $\,P(y) = Q(y) + T(y) \cdot \prod_{i=0}^3 (y-y_i)\,$ for some arbitrary polynomial $\,T(y)\,$.