Show that every greatest common divisor is a nonzero constant multiples of d(x)

Question

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?

Answer 1

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 $k\neq 0$.