Polynomial Division

Theorem: Polynomial Division

Let $R$ be a commutative ring and let $A(x)$ and $B(x)$ be two univariate polynomials over $R$ such that the degree of $A$ is greater than or equal to the degree of $B$.

If $B(x)$ is nonzero, then there exist unique polynomials $Q(x)$ and $R(x)$ such that

$$A(x)=Q(x)B(x)+R(x),$$

where

• $\mathrm{deg}(Q)=\mathrm{deg}(A)-\mathrm{deg}(B)$
• $\mathrm{deg}(R)<\mathrm{deg}(B)$ or $R(x)$ is the zero polynomial.

We call $A$ the dividend, $B$ the divisor, $Q$ the quotient and $R$ the remainder.

PROOF

TODO

Definition: Divisibility

If $R(x)=0$, then we say that $A$ is divisible by $B$.

Polynomial Remainder Theorem (Little Bézout's Theorem)

The remainder $R$ upon the division of a polynomial $A(x)$ with a polynomial $B(x)=x-p$ is the value of $A$ at $p$.

$$R=A(p)$$

PROOF

The polynomial division theorem tells us that

$$A(x)=(x-p)Q(x)+R(x)$$

Since the degree of $B(x)$ is $1$, the degree of $R$ must either be $1$, i.e. $R$ is a constant and contains no variables or $R$ must be zero. We can therefore denote $R(x)$ as just $R$. For $x=p$ we obtain

$$A(p)=(p-p)Q(p)+R=0+R=R$$