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Theorem 1 (Integer Long Division)   For every pair of integers $a, b\neq 0$ there exist unique integers $q$ and $r$ such that:

1. $a=b\cdot q + r,$
2. $0\leq r < |b|$ .

Example 1   Let $a=10$ and $b=-3$ . Then $q=-3$ and $r=1$ correspond to the long division: $$10=(-3)\cdot(-3)+1.$$

Definition 1   The number $r$ as in the theorem is called the remainder of the division of $a$ by $b$ . The numbers $a,\ b$ and $q$ are called the dividend, divisor and quotient respectively.

Theorem 2 (Polynomial Long Division)   Let $R$ be a commutative ring with non-zero unity and let $a(x)$ and $b(x)$ be two polynomials in $R[x]$ , where the leading coefficient of $b(x)$ is a unit of $R$ . Then there exist unique polynomials $q(x)$ and $r(x)$ in $R[x]$ such that:

1. $a(x)=b(x)\cdot q(x) + r(x),$
2. $0\leq \deg(r(x)) < \deg b(x)$ or $r(x)=0$ .

Example 2   Let $R=\Ints$ and let $a(x)=x^3+3$ , $b(x)=x^2+1$ . Then $q(x)=x$ and $r(x)=-x+3$ , so that: $$x^3+3=x(x^2+1)-x+3.$$

Example 3   The theorem is not true in general if the leading coefficient of $b(x)$ is not a unit. For example, if $a(x)=x^3+3$ and $b(x)=3x^2+1$ then there are no $q(x)$ and $r(x)$ with coefficients in $\Ints$ with the required properties.

long division is owned by alozano.