Polynomial division with remainder. GCD and LCM of polynomials

Just like with integers, we can divide one polynomial by another, and there will usually be a remainder. For polynomials $f(x)$ and $g(x)$ (with $g(x) \neq 0$), there exist unique polynomials $q(x)$ (the quotient) and $r(x)$ (the remainder) such that:

where the degree of $r(x)$ is less than the degree of $g(x)$. The greatest common divisor (GCD) of two polynomials is the highest-degree polynomial that divides both without remainder. The least common multiple (LCM) is the lowest-degree polynomial that both polynomials divide.

Important properties

• The remainder $r(x)$ has degree less than $g(x)$ or is zero.

• The quotient and remainder are unique for given $f(x)$ and $g(x)$.

• The GCD of two polynomials can be found using the Euclidean algorithm, similar to integers.

• Any two polynomials have a GCD and an LCM (up to multiplication by a nonzero constant).