Let Z[x] be the ring of all polynomials with integer coefficients. Let I = {f(x) in Z[x]| f(2) = 0}. Prove that I is an ideal.

I know the definition: A subring A of a ring R is called a (two-sided) ideal of R if for every r in R and every a in A ra and ar are in A.

There is also an Ideal Test: A nonempty subset A of a ring R is an ideal of R if 1. (a-b) belongs to A whenever a, b belong to A. 2. ra and ar are in A whenever a is in A and r is in R.

I don't think this should be a hard problem at all, but we haven't really had any examples in class and I am not sure how to apply the test.