Polynomial regression

A polynomial regression wants to model a non-linear, polynomial relationship within the data. It can be just reduced to a linear regression via variable substitution, reducing the features to linear ones. A generic polynomial model is
y(x)=i=0i=maixi=a0+a1x+a2x2++amxmy(x) = \sum_{i=0}^{i=m}a_i x^i = a_0 + a_1x + a_2x^2 + \ldots + a_mx^m
The problem can be reduced to that of a multiple linear regression where the features are the polynomials of
{x1=xx2=x2xm=xm\begin{cases} x_1 = x \\ x_2 = x^2 \\ \ldots \\ x_m = x^m \end{cases}
so that the model is linear in all of these new features.
The same applies to non-polynomial features, say exponentials or other functions.