The polynomials are non-negative over the interval $[0,1]$.

The set of polynomials of degree $n$ is symmetric with respect to $t=1/2$.

Each polynomial has only one maximum over the interval $[0,1]$ at $t=\frac{i}{n}$.

The set of polynomials of degree $n$ forms a partition of unity.

A polynomial can always be written as a linear combination of polynomials of higher degree.