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Suppose $F$ is a field, let $f(x), g(x), h(x) \in F[x]$ and $d(x)=\gcd(f(x),g(x))$. If $f(x) \mid h(x)$ and $g(x) \mid h(x)$, prove that $f(x)g(x) \mid d(x)h(x)$.

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    $\begingroup$ What have you tried? Where are you stuck? $\endgroup$ – Robert Shore Oct 9 '19 at 15:43
  • $\begingroup$ $f,g\mid h\iff {\rm lcm}(f,g)\mid h,\,$ and $\,{\rm lcm}(f,g) = fg/\gcd(f,g).\,$ Most of the proofs for integers in the linked dupes also work here since $F[x]$ is a PID so enjoys GCDs, unique factorization, Bezout identity etc. $\endgroup$ – Bill Dubuque Oct 9 '19 at 15:50
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$F[X]$ is a principal domain, so $d(x)=a(x)f(x)+b(x)g(x)$

Write $h(x)=u(x)f(x)=v(x)g(x)$.

$d(x)h(x)=$

$(a(x)f(x)+b(x)g(x))h(x)=$

$a(x)v(x)f(x)g(x)+b(x)u(x)f(x)g(x)$

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