6.34.7 Euclidean quotient and euclidean remainder: quorem
The quorem command finds the quotient
and remainder of the Euclidean division of two polynomials
(see also Section 6.5.10 and Section 6.28.4).
-
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 the Euclidean
division of P and Q.
Example.
Input:
quorem((x^3+x^2+1)%13,(2*x^2+4)%13)
Output:
⎡
⎣ | ⎛
⎝ | ⎛
⎝ | −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 mod13.