A polynomial p(x) divides a polynomial g(x) if these exists a polynomial q(x) such that g(x) = p(x) q(x).

Theorem (The Division Algorithm for Polynomials): Let p(x) be a polynomial of degree n, and let g(x) be a polynomial of degree \( m \ge 0 . \) Then there exist unique polynomials q(x) and r(x) such that
\[ p(x) = q(x)\,g(x) + r(x) , \]
where the degree of r(x) is less than m. ■