Dividing a polynomial by x - a works like long division of numbers: a quotient and a remainder. The remainder theorem says the remainder is p(a), so x - a is a factor exactly when p(a) = 0. Polynomial identities like a² - b² = (a + b)(a - b) and (a + b)² = a² + 2ab + b² are proved by expanding, and they let you rewrite an expression by reading its structure.