What is the maximum degree polynomial passing the points below?? $$(0,2) (1,4) (2,8) (3,14) (4,22)$$
I guess the answer is 2 and the corresponding polynomial is $x^2+x+2$, but how can I prove that there is no polynomial of degree $3$ or $2$ passing these points?
If $p$ is any polynomial passing through these points then $p(x)+x(x-1)(x-2)(x-3)(x-4)q(x)$ also passes through these points for any polynomial $q$. So there is no maximum value for the degree of the polynomial.