Let $a(x), b(x), d(x)$ be polynomials

I need to show that every greatest common divisor $d(x)$ of $a(x)$ and $b(x)$ is a nonzero constant multiples of $d(x)$

I know it should be easy but i’m stuck, so any ideas?

  • $\begingroup$ Oh sry, It’s polynomials $\endgroup$ – user32091 Nov 23 '17 at 0:18

Assume $d$ and $e$ are the gcd's of $a$ and $b$. Then by definition $e|d$ and $d|e$. Thus $e=kd$ for some polynomial $k$. Since $e|d$ the degree of $e$ is less than or equal to the degree of $d$, so $k$ must have degree $\leq 0$. Therefore $k$ is a constant, since $0 \neq e = kd$, we must have $k\neq 0$.


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