A polynomial with integer coeffiecients is an expression of the form $\displaystyle f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ where $\displaystyle a_i\in\mathbb{Z}$. If $\displaystyle a_n\neq0$, $\displaystyle f(x)$ is said to be of degree $\displaystyle n$. Denoting the set of all such polynomials as $\displaystyle \mathbb{Z}[x]$, show that $\displaystyle \mathbb{Z}[x]$ is a ring with respect to a natural definition of addition and multiplication of polynomials.

So, Let $\displaystyle f_1(x),f_2(x)\in\mathbb{Z}[x]$, where $\displaystyle f_1(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ and

$\displaystyle f_2(x)=b_kx^k+b_{k-1}x^{k-1}+...+b_1x+b_0$. Now we have

$\displaystyle f_1+f_2=(a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0)+(b_kx^k+b_{k-1}x^{k-1}+...+b_1x+b_0)=$.

Clearly, anwhere $\displaystyle n=k$, the coefficient will be $\displaystyle a_n+b_k$, which is an integer. All of the rest of the terms will remain unchanged and so will necessarily have integer coefficients, therefore $\displaystyle f_1+f_2\in\mathbb{Z}[x]$ and we conclude that the sum of two functions in $\displaystyle \mathbb{Z}[x]$ is also in $\displaystyle \mathbb{Z}[x]$.

OK. I'm trying here but I'm stuck. Help?