6:58 not going to lie, I spent an inordinate amount of time trying to figure out how it could possibly be O(n)! before realizing the exclamation was only punctuation, not a factorial

13:00 for those of you who are wondering why he is plugging in the square of x_i. It's because when we use the coefficient form, if we just plug in x_i, it would be interpreted as powers 0,1,2,...., so if we plug in x^2 it would be interpreted as (x^2)^0. (x^2)^1, (x^2)^2, (x^2)^3, which are powers 0,2,4,6...which are even terms.

An aside: When you mention that any polynomial of degree N can be uniquely determined by N+1 points? That's how many forms of forward error correction work, suitably modified for modular arithmetic. If you want to send your message you break it up into N parts, each of which becomes a point on a polynomial of degree N-1. Then generate as many extra points on the polynomial as you need to generate a list of K points where K>N, and transmit those K points through your unreliable communications channel. As long as at least N of the K make it through to the other end, you can recover the original message.... and thus your phone call can keep on going even with a little radio interference taking place. You can tune K to be optimal for the reliability of your communications channel and the acceptable level of overhead.

26:44 friend, my mind has been blown since 6:00

I like the fact you actually laid out the pseudocode for the algorithm at the end of each section, most channels that cover topics like these usually leave it at the block diagram. Shows you really know your stuff.

An amazing breakdown of the FFT!

Who came here from 3blue1brown and fell in love with this channel?

This explanation is insane!