finding the order of the polynomials in a polynomial groups under addition My question is:   if  G:= Group of polynomials under addition with elements from $\{0,1,2,3,4,5,6,7,8,9\}$
$f(x)=7x^2+5^x+4$ and $g(x)=4x^2+8^x+6$  ,$f(x)+g(x)=x^2+3x$
what is the order of $f(x)$,$g(x)$,$f(x)+g(x)$
i find that the answers are $10$,$5$ and $10$ . Am i correct??
(i noticed that zero polynomial is the identity here)
then if $h(x)=a_nx^n+...+a_0$ belongs to $G$ what is the order of it? given that

$gcd(a_1,a_2,,,a_n)=1$,

$gcd(a_1,a_2,,,,a_n)=2$,

$gcd(a_1,a_2,,,a_n)=5$,

$gcd(a_1,a_2,,,a_n)=10$

Here i do not understand how to proceeded ?
Find the solution in the attached picture
Find the solution in the attached picture