A polynomial with integer coeffiecients is an expression of the form where . If , is said to be of degree . Denoting the set of all such polynomials as , show that is a ring with respect to a natural definition of addition and multiplication of polynomials.

So, Let , where and

. Now we have

.

Clearly, anwhere , the coefficient will be , which is an integer. All of the rest of the terms will remain unchanged and so will necessarily have integer coefficients, therefore and we conclude that the sum of two functions in is also in .

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