Complete Polynomial Factorization

Understanding Complete Factorization

We have learned to factor polynomials, for example using the Factor Theorem. However, sometimes the factorization result still leaves factors that are not linear (like quadratic factors) which cannot be factored further using real numbers.

Complete Factorization (or Complete Linear Factorization) is the process of factoring a polynomial into a product of linear factors, where these factors may involve complex numbers.

This concept is based on the Fundamental Theorem of Algebra, which states that every polynomial of degree $n \ge 1$ has exactly $n$ roots (zeros) in the set of complex numbers (including real roots and repeated roots).