Consider two continuous real valued functions $f$, and $g$ over the interval $[a,b]$. For a given $\epsilon > 0$, for each function the we can use the Weierstrass approximation theorem to express the function as polynomials $f \approx \mathcal{P}_f$ and $g \approx \mathcal{P}_g$, such that $\lvert f - \mathcal{P}_f\lvert < \epsilon$ and $\lvert g - \mathcal{P}_g\lvert < \epsilon$, over the interval $[a,b]$. Is there anything we can say about the approximating the product of the two functions $f \cdot g$, i.e., $\mathcal{P}_{f\cdot g}$ where $\lvert f\cdot g - \mathcal{P}_{f\cdot g}\rvert < \epsilon$ on the interval $[a,b]$. In other words, is there any relation between $\mathcal{P}_{f\cdot g}$, $\mathcal{P}_f$, and $\mathcal{P}_g$?
In particular, I am interested if from $\deg(\mathcal{P}_f)$ and $\deg(\mathcal{P}_g)$ we can say anything about $\deg(\mathcal{P}_{f\cdot g})$. I can appreciate that $\deg(\mathcal{P}_f) + \deg(\mathcal{P}_g)$ may be greater than $\deg(\mathcal{P}_{f\cdot g})$; as the mutliplication may conspire to cancel out higher terms (e.g., $g \ll f$, such that $\mathcal{P}_f$ has ''overfit'' $f$ to achieve $\epsilon$ in the composition.) However, I cannot see if one can guarantee that this is the upper limit, i.e., $$\deg(\mathcal{P}_{f\cdot g}) \leq \deg(\mathcal{P}_f) + \deg(\mathcal{P}_g)?$$
An example that the component degree sum is larger can be seen from $f(x) = 1+x$, and $g(x) = (1+x)^{-1}$. For small enough $\epsilon$, both have approximating polynomials greater than degree zero, while the product's approximating polynomial is trivially degree zero.
Is there a counter-example to the suggestion inequality, i.e., where
$$\deg(\mathcal{P}_{f\cdot g}) > \deg(\mathcal{P}_f) + \deg(\mathcal{P}_g)?$$
Disclaimer: This is a reformulation of a question I mistakenly asked about function composition.
