# long division

In this entry we treat two cases of long division.

## 1 Integers

###### Theorem 1 (Integer Long Division).

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

1. 1.

$a=b\cdot q+r,$

2. 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.

## 2 Polynomials

###### 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. 1.

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

2. 2.

$0\leq\deg(r(x))<\deg b(x)$ 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.

 Title long division Canonical name LongDivision Date of creation 2013-03-22 15:09:07 Last modified on 2013-03-22 15:09:07 Owner alozano (2414) Last modified by alozano (2414) Numerical id 7 Author alozano (2414) Entry type Theorem Classification msc 12E99 Classification msc 00A05 Classification msc 11A05 Synonym division algorithm Related topic Polynomial Related topic PolynomialLongDivision Related topic MixedFraction Defines dividend Defines remainder