11.8.7 Euclidean quotient and euclidean remainder
The quorem command finds
the quotient and remainder of the Euclidean division of two polynomials
(see also Section 7.1.10 and Section 11.2.4).
-
quorem takes two mandatory arguments and one optional
argument:
-
P and Q, two polynomials with
coefficients in ℤ/pℤ.
- Optionally x, the variable (by default
x), if P and Q are given as expressions.
- quorem(P,Q ⟨,x⟩)
returns the list of the quotient and remainder of dividing P by Q.
Example
| quorem((x^3+x^2+1)%13,(2*x^2+4)%13) |
|
| |
| ⎡
⎣ | ⎛
⎝ | ⎛
⎝ | −6 | ⎞
⎠ | %13 | ⎞
⎠ | x+ | ⎛
⎝ | −6 | ⎞
⎠ | %13, | ⎛
⎝ | ⎛
⎝ | −2 | ⎞
⎠ | %13 | ⎞
⎠ | x+ | ⎛
⎝ | −1 | ⎞
⎠ | %13 | ⎤
⎦ |
| | | | | | | | | | |
|
Indeed,
x3+x2+1=(2x2+4)·x+1/2+5x−4/4
and −3· 4=−6· 2≡ 1(mod 13 ).