$\gcd$ of polynomials over a field I have the polynomials $f,g\neq 0 $ over a field $F$. We know also that $\gcd(f,g)=1$ and 
$$
       \det \begin{pmatrix}
        a & b \\
        c & d \\
        \end{pmatrix}\neq 0.
$$
I need to prove that $\gcd(af+bg,cf+dg) = 1 $ for every $a,b,c,d \in F$. 
I really do not know how to start answer the question. Thanks for helpers!

Just work through the equations of gcd (bearing in mind that you're working in $f(x)$, hence constants do not matter):
$ \gcd(af+bg, cf+dg) = \gcd(adf+bdg, cbf + bdg) = \gcd( (ad-bc)f, cbf + bdg) = \gcd( f, cbf+bdg) = \gcd(f, bdg) = \gcd(f, g) = 1$
