# 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 ?