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[parent] long division (Theorem)

In this entry we treat two cases of long division.

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.

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 $R=\Ints$ 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.
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See Also: polynomial, polynomial long division

Other names:  division algorithm
Also defines:  dividend, remainder

This object's parent.

Attachments:
proof of long division (Proof) by Thomas Heye
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Cross-references: properties, coefficients, unit, leading coefficient, polynomials, non-zero unity, commutative ring, quotient, divisor, division, theorem, number, integers
There are 72 references to this entry.

This is version 4 of long division, born on 2005-03-22, modified 2006-02-21.
Object id is 6899, canonical name is LongDivision.
Accessed 11931 times total.

Classification:
AMS MSC00A05 (General :: General and miscellaneous specific topics :: General mathematics)
 12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)
 11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)

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