Compute the greatest common divisor of two polynomials over QQ

Compute the greatest common divisor of p(x) = x^4 + 3x^3 + 10x^2 + 18x + 24 and q(x) = x^4 + 2x^3 + 13x^2 + 12x + 42 in \mathbb{Q}[x].


We will carry out the Euclidean algorithm. Note that p(x) = q(x) + (x^3-3x^2+6x-18), q(x) = (x^3-3x^2+6x-18)(x+5) + (22x^2+132), and x^3-3x^2+6x-18 = (22x^2+132)(\frac{1}{22}x - \frac{3}{22}). In particular, 22x^2+132 is a greatest common divisor of p(x) and q(x) in \mathbb{Q}[x]. Since 22 is a unit, we see that x^2 + 6 is also a greatest common divisor.

Indeed, p(x) = (x^2+6)(x^2+3x+4) and q(x) = (x^2+6)(x^2+2x+7). Note that x^2+3x+4 is irreducible in \mathbb{Z}[x] (hence \mathbb{Q}[x]) since \{a+b=3, ab=4 \} has no integer solutiions, and similarly x^2+2x+7 is irreducible, and these are not associates.

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