GCF of two large polynomials

Question

What is the Greatest Common Factor for $P(x) = x^{50} - 2x^2 - 1$ and $Q(x) = x^{48} - 3x^2 - 4$?

I tried long division, but even the second step is too complicated that I gave up.

I feel there's more to it than the usual Euclidean Algorithm or long division (like the fact that the degree of the first term is drastically bigger than the second term), but I can only draw blanks.

Anyone want to help me shed some light on this particular problem?

Answer 1

Notice that if $f$ is the GCD, then $f \mid \gcd (P, x^2 Q)$, so that

$$f \mid \gcd (P, P-x^2 Q) = \gcd (P, 3x^4 +2x^2 - 1) ,$$

which implies

$f \mid 3x^4 + 2x^2 -1 = 3 (x - \Bbb i) (x + \Bbb i) (x - \frac 1 {\sqrt 3}) (x + \frac 1 {\sqrt 3}) .$

Notice that $Q(\Bbb i) = 1 + 3 - 4 = 0 = Q(-\Bbb i)$, and that $Q(\frac 1 {\sqrt 3}) = Q(-\frac 1 {\sqrt 3}) \ne 0$ (obviously). This means that you only have to retain the part $(x+ \Bbb i) (x - \Bbb i) = x^2 + 1$, the other two "parasitic" factors having been introduced by the $x^2$ in $x^2 Q$. This shows that $f = x^2 + 1$.

Answer 2

It is also a common factor of $x^2Q(x)-P(x)$.
The smaller polynomial is easier to factor, so you can find all its roots.
Check whether $P(r)=0$ and $Q(r)=0$ for each root $r$.
Then construct the common factor polynomial from the list of common roots.