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≠0 there exist unique integers q and r such that:
-
1.
a=b⋅q+r,
-
2.
0≤r<|b|.
Example 1.
Let a=10 and b=-3. Then q=-3 and r=1 correspond to the long division:
10=(-3)⋅(-3)+1. |
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)⋅q(x)+r(x),
-
2.
0≤deg(r(x))<degb(x) or r(x)=0.
Example 2.
Let R=ℤ and let a(x)=x3+3, b(x)=x2+1. Then q(x)=x and r(x)=-x+3, so that:
x3+3=x(x2+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)=x3+3 and b(x)=3x2+1 then there are no q(x) and r(x) with coefficients in 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 |