Can someone please explain to me why $y_0 (x) = 1$ when $\alpha = 0$ and $y_1 (x) = x$ when $\alpha = 1$... $y_0 (x) = 1 - 2x^2$ when $\alpha = 2$ and so on. I understand that subbing in those values creates a $0$ factor. But how do they simplify to those polynomials? Wouldn't it just be the sum of $0$'s? ie $\sum 0$?