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.

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