Division with Remainder (Polynomials)

theorempolynomials

Intuition

Similarly to how we have division with remainder in other rings ($\mathbb{Z},\mathbb{Z}[\sqrt{2}]$, etc.), we have unique division in the ring of polynomials.

Formal Statement

For polynomials $P,Q\in R[x]$, $Q\ne 0$. Assume either

1. $R=F$ is a field, or
2. $Q(x)$ monic ($LT(Q)=1$), Then there exists $D(x),r(x)\in R[x]$ such that $P(x)=D(x)\cdot Q(x)+r(x)$ and $\mathrm{deg}r(x)<\mathrm{deg}Q(x)$.

Corollaries

1. In $F[x]$ we have Euclid’s algorithm, we have $\gcd (P(x),Q(x))$ for any $P,Q$ not both $0$, computable with Euclid’s algorithm, and $\exists u,v\in F[x]$ such that $u\cdot P+v\cdot Q=\gcd (P,Q)$.
2. If $I⊴F[x]$, then $I=(0)$ or $I=(P(x))$, $P(x)$ is the unique smallest nonnegative degree monic element in $I$. (We say $F[x]$ is an example of a P.I.D. (principal ideal domain)).