The Division Algorithm for Polynomials over a Field

The Division Algorithm for Polynomials over a Field

Recall from the Roots of Polynomials over a Field page that if $(F, +, \cdot)$ is a field and $f \in F[x]$ then $c \in F$ is said to be a root of $f$ is:

\begin{align} \quad f(c) = 0 \end{align}

We looked at two important results. The first being that if $f(x) \neq 0$ then $c \in F$ is a root of $f$ if and only if $(x - c) | f(x)$.

The second result stated that if $f(x) \neq 0$ and $\deg (f) = n$ then $f$ has at most $n$ distinct roots.

We now state a very important algorithm called the division algorithm for polynomials over a field. The result is analogous to the division algorithm for natural numbers.

Theorem 1 (The Division Algorithm for Polynomials over a Field): Let $(F, +, \cdot)$ be a field and let $f, g \in F[x]$ with $g(x) \neq 0$. Then there exists unique $q, r \in F[x]$ such that $f(x) = g(x)q(x) + r(x)$ with the property that either $r(x) = 0$ or $\deg(r) < \deg(g)$.

For example, consider the field of real numbers and the polynomials $f(x) = x^3 - 2x^2 + x + 3$ and $g(x) = 2x^2 - 1$. Then:

\begin{align} \quad \underbrace{x^3 - 2x^2 + x + 3}_{f(x)} = \underbrace{(2x^2 - 1)}_{g(x)} \underbrace{\left ( \frac{1}{2}x + 1 \right )}_{q(x)} + \underbrace{\left ( \frac{3}{2}x + 2 \right )}_{r(x)} \end{align}

We see that $\deg (r) < \deg(g)$. The polynomials $q(x)$ and $r(x)$ can be obtained by polynomial long-division.

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