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 $f(x)g(x) \mid d(x)h(x)$.
2026-03-31 22:42:52.1774996972
Divisibility with GCD
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$F[X]$ is a principal domain, so $d(x)=a(x)f(x)+b(x)g(x)$
Write $h(x)=u(x)f(x)=v(x)g(x)$.
$d(x)h(x)=$
$(a(x)f(x)+b(x)g(x))h(x)=$
$a(x)v(x)f(x)g(x)+b(x)u(x)f(x)g(x)$