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:

  x−1

• 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:

  ⎡ ⎣

  1,2

  ⎤ ⎦

  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:

  rem

  ⎛ ⎝

  x3−1,x2−1

  ⎞ ⎠

• Input (in Maple mode):
  Rem(x^3+3*x,2*x^2+6*x+5) mod 5
  Output:

  2x

  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:

  12x

If Xcas is not in Maple mode, polynomial division in ℤ/pℤ[X] is entered, for example, by: