Where does this given info come from?

Question

I am following this slide on FFT. On the last page, it says:

I would like to ask where $a = A(2)$ comes from. Thanks.

Answer 1

In base two, a number $b = b_{n-1}\dots b_0$ stands for $b = \sum_{i=0}^{n-1} b_i 2^i$.

Defining the polynomial $B(X) =\sum_{i=0}^{n-1} b_i X^i$, you have $b = B(2)$.

Note : in base $\beta$, the number $b = b_{n-1}\dots b_0$ is $B(\beta)$.