Division Algorithm

Division algorithm. Given polynomials $f(x)$ and a nonzero $g(x)$, there exist unique polynomials $q(x)$ (the quotient) and $r(x)$ (the remainder) with

If $r(x)\equiv0$, then $g(x)$ and $q(x)$ are factors of $f(x)$.

Special case $g(x)=x-a$. Here $\deg r<1$, so $r(x)$ is just a constant $c$: $f(x)=(x-a)q(x)+c$. Substituting $x=a$ gives $c=f(a)$.

Remainder Theorem. If $f(x)$ is divided by $x-a$, the remainder is $f(a)$. Hence $f(a)=0$ if and only if $x-a$ is a factor of $f(x)$ -- this is the Factor Theorem.

Zeros and multiplicity. $a$ is a zero of $f(x)$ if $f(a)=0$; then $x-a$ is a factor. If $f(x)=(x-a)^k g(x)$ with $g(a)\ne0$, the exponent $k$ is the zero's multiplicity, and $k$ can never exceed $\deg f$.