6.28.3 Euclidean remainder: rem Rem
The rem command finds the remainder of the Euclidean division
of two polynomials.
-
rem takes two mandatory arguments and one optional
argument:
-
P and Q, two polynomials.
- Optionally x, the variable (by default
x), if P and Q are given as expressions.
- rem(P,Q ⟨ ,x⟩) returns
the Euclidean remainder of P divided by Q.
Examples.
-
Input:
rem(x^3-1,x^2-1)
Output:
- To have the remainder of x2+2x+4 by x2+x+2, you can also do:
Input:
rem([1,2,4],[1,1,2])
Output:
i.e. the polynomial x+2.
Rem is the inert form of rem; namely, it evaluates to
rem for later evaluation. It is used when Xcas is in
Maple mode (see Section 3.5.2) to compute the euclidean remainder
of the division of two polynomials with coefficients in
ℤ/pℤ using Maple-like syntax.
Examples.
-
Input (in Xcas mode):
Rem(x^3-1,x^2-1)
Output:
- Input (in Maple mode):
Rem(x^3+3*x,2*x^2+6*x+5) mod 5
Output:
This division was done using modular arithmetic, unlike with
rem(x^3+3*x,2*x^2+6*x+5) mod 5
where the division is done in ℤ[X] and reduced after to:
If Xcas is not in Maple mode, polynomial division
in ℤ/pℤ[X] is entered, for example, by:
rem((x^3+3*x)% 5,(2x^2+6x+5)%5)