How do different types of polynomial division relate to each other?

Question

How does factoring relate to long division?

How does long division relate to synthetic division?

How does synthetic division relate to factoring?

Answer 1

Interesting question. They're all related to factoring. When you have a polynomial, one of the main goals is to find its roots and the only way to do it is by factoring. Long division is just a method of factoring a polynomial if you happen to know one of its factors. Synthetic division is just a helpful diagrammatic shortcut of doing long division with polynomials.