Examples: Polynomial Factorization

This lecture presents examples of polynomial factorization, starting with the factorization of z^2 - 2z + 1 as (z - 1)^2, followed by the factorization of z^2 - 1 as (z + 1)(z - 1). The decomposition of z^3 - 1 in C is discussed, showing the roots as z = 1, z = -1/2 - i√3/2, and z = -1/2 + i√3/2. The factorization of z^4 - 1 is demonstrated using the identity (z^2)^2 = (z^2 + 1)(z^2 - 1), leading to the factors (z - 1)(z^2 + z + 1). The lecture also covers polynomial division, illustrating the division of z^3 - 1 by (z - 1) to obtain z^2 + z + 1.