How do we go about factorizing boolean expressions?

Question

How do we know how to go about factorizing a boolean expression when there are so many ways?

For example, the factorized form of $ABC + A'B'C'$ is $(A + C')(B' + C)(A' + B)$, but how do we know how to derive this result?

Answer 1

$ABC+A'B\,'C'$ is a sum of products form or, if you’re thinking of it in logical terms, a disjunctive normal form; $(A + C')(B' + C)(A' + B)$ is a product of sums form or, in logical terms, a conjunctive normal form. There are algorithms for converting Boolean expressions to product of sums form and logical expressions to conjunctive normal form; you might start with the Wikipedia articles on canonical forms in Boolean algebras and conjunctive normal form.