Polynomial Division

Dividing polynomials $p(x)$ by $d(x)$ gives two unique polynomials, the quotient $q(x)$ and remainder $r(x)$, such that

\[p(x) = q(x)d(x) + r(x),\]

and the degree of $r(x)$ is less than the degree of $d(x)$.

Remainder and Factor Theorems

If $p(x)$ has a degree of at least 1, and $d(x)$ has degree 1, then $r(x)$ is constant.

The Remainder theorem says that if $k$ is the root of $d(x)$, then the remainder $r(x) = p(k)$.

\[p(x) = q(x)\cdot(x-k) + r(x) \implies p(k) = r(k)\]

The Factor Theorem says that $k$ is a root of $p(x)$ if and only if $d(x)$ is a factor of $p(x)$, that is when $r(x) = 0$.