Why are polynomials of degree > 3 not preferred for representing curves in practice?

Question

I know that in practice, polynomials higher than cubics are not preferred for representing curves. What are the reasons for this ? Is it just because solving the inverse problem of finding $x$ given value of $P(x)$ is difficult for higher-degree polynomials ?

Answer 1

We know that a degree $n$ polynomial can be used to interpolate $n+1$ points perfectly.

However, a high degree polynomial tend to fluctuate a lot and in the context of regression, we might be just learning the noise rather than the underlying behavior of the data pattern of interest. We call such behavior overfitting.

However, it is not clear that degree $3$ is the default answer, in fact, there shouldn't be such an answer. It is highly depending on the data. If we use a very high degree, we are prompt to overfitting, if we use a model that is too simple, we are prompt to underfitting.

Runge's phenomenon might be of interest to you too.