The unicity of $P$ and $g$ in the factorization of the function $f$

The unicity of $P$ and $g$ in the factorization of the function $f$

The motivation to this question can be found in: Examples of real analytic
functions with finite number of zeros that are not polynomials
My quetion is concerned with the unicity of $P$ and $g$ in the
factorization of the function $f$.