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Here is my current thought; Let $f = df', g = dg'$ for some polynomials $f',g'$

by division theorem

$f = gq + r$ for some polynomial $r,q$. We have $r = f - gq = df' - dg'q = d(f'-dg')$; hence, $d$ divides $r$; similarly, $d'$ divides $r$.

So theree are polynomial $a,b$ such that $r = da = d'b$.

My plan is to show that $r=gcd(f,g)$ as well, so I have degree($r$) = degree($da$) = degree($d'b$) = degree($d$) = degree($d'$) implies that $a,b$ just constant. I can write $d = \frac{b}{a}d'$

But I'm stuck at how to prove $r=gcd(f,g)$ Any Ideas ?

ElleryL
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1 Answers1

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This follows directly from the definition of a greatest common divisor, which is that it is a common divisor, and every common divisor divides it. Thus as $d'(x)$ is a common divisor, and $d(x)$ is a greatest common divisor, $d'(x)|d(x)$, which is equivalent to saying that there exists a $c(x)$ such that $d'(x)c(x) = d(x)$.

aefrrs
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