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,b0 there exist unique integers q and r such that:

  1. 1.


  2. 2.


Example 1.

Let a=10 and b=-3. Then q=-3 and r=1 correspond to the long division:

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, divisorMathworldPlanetmathPlanetmath 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 polynomialsPlanetmathPlanetmath 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.


  2. 2.

    0deg(r(x))<degb(x) or r(x)=0.

Example 2.

Let R= and let a(x)=x3+3, b(x)=x2+1. Then q(x)=x and r(x)=-x+3, so that:

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)=x3+3 and b(x)=3x2+1 then there are no q(x) and r(x) with coefficients in 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