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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 < \vert b\vert$.
Example 1   Let $ a=10$ and $ b=-3$. Then $ q=-3$ and $ r=1$ correspond to the long division:
$\displaystyle 10=(-3)\cdot(-3)+1.$
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=\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:
$\displaystyle x^3+3=x(x^2+1)-x+3.$
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.



"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, integers
There are 54 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 7921 times total.

Classification:
AMS MSC: 00A05 (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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