Let h be a hcf of $f, g \in K[x]$ Then there exists polynomials a and b such that
$h = af + bg$
Can anyone explain this theorem to me intuitively?
2026-04-09 00:24:06.1775694246
Highest common factors of polynomials
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Let $h$ be a nonzero polynomial of the form $af+bg$ with smallest possible degree. If you divide either $f$ or $g$ by $h$ then the remainder is also of that form, and has lower degree, so it must be zero.
So $h$ is a common factor of $f$ and $g$. But any common factor of $f$ and $g$ divides $h$, since $h$ is of the form $af+bg$.