show 1 is not a linear combination of the polynomials 2 and x in Z[x] I want to prove that 1 is not a linear combination of the polynomials 2 and x in Z[x], can you give any hints for this? 
What I have done is construct $f(x), g(x) \in Z[x]$ such that 
\begin{align}
2f(x) + x g(x)=1
\end{align}
Let 
\begin{align}
f(x) = \sum_{n=0}^k a_n x^n, \quad g(x) = \sum_{m=0}^l b_m x^m
\end{align}
where $a_n, b_n \in Z$, collecting coefficient terms, I see that 
\begin{align}
2 a_0 = 1
\end{align}
and this is impossible since $a_0 \in Z$. 
I want to know some other proof
Are there any other proofs? 

Indeed, there are no polynomials $f,g$ with integer coefficients such that $xf(x)+2g(x)=1$,
since the constant coefficient on the left-hand side is divisible by 2.
