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Homelong division

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

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.

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

2. $0\leq\deg(r(x))<\deg b(x)$ or $r(x)=0$.

###### Example 2.

Let $R=\mathbb{Z}$ 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.

## Mathematics Subject Classification

12E99*no label found*00A05

*no label found*11A05

*no label found*

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