Euclidean quotient and euclidean remainder: quorem

### 5.34.7 Euclidean quotient and euclidean remainder: quorem

The quorem command finds the quotient and remainder of the Euclidean division of two polynomials (see also Section 5.5.10 and Section 5.28.4).

- 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.