long division

In this entry we treat two cases of long division.

1 Integers

Theorem 1 (Integer Long Division).

For every pair of integers $a\mathrm{,}b\mathrm{\ne }\mathrm{0}$ there exist unique integers $q$ and $r$ such that:

1. $a=b\cdot q+r,$

2. .

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 $a\mathrm{,}b$ and $q$ are called the dividend, divisor and quotient respectively.

2 Polynomials

Theorem 2 (Polynomial Long Division).

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

1. $a(x)=b(x)\cdot q(x)+r(x),$

2. or $r(x)=0$.

Example 2.

Let $R=\mathbb{Z}$ 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 $\mathbb{Z}$ with the required properties.