Increasing degree for Newton Interpolation

Question

Given $n+1$ nodes $x_0,...x_n$ and a polynomial of degree $n$ called $p_n$ with $p_i=f_i$ for $i=1,...,n$. We looked into what happens when we increase the degree of $p_n$ to $m:=n+1$. It would be easy if $$p_m(x)=p_n(x)+g_m(x) $$ Now we know that that $g_m(x_i)=p_m(x_i)-p_n(x_i) = f(x_i)-f(x_i)=0$ for $i=1,...,n$. The notes says it follows that $$ g_m(x)=a_m(x-x_0)\cdot...\cdot(x-x_n) $$ Now I want to understand how to get to this last conclusion. Obviously if we have $n$ roots $x_0,...x_n$ give for $p_m$ than these roots can be written as linearfactors. But this seems not very convincing and I wanted to ask if there is a way to show this using the Uniqueness of the interpolation polynomial?

Answer 1

$g_m$ is a polynomial of degree $n+1$ and by construction you know its $n+1$ roots at $x_0,x_1,...,x_n$. Then it is trivial that $g_m$ must be the product of the linear factors of the roots with a constant coefficient, as claimed.