6. List all the polynomials of degree 2 in Zz[x]. Which of these are equal as functions from Z2 to Zz? 7. Find two distinct cubic polynomials over Z2 that determine the same function from Zz to Z2. 8. For any positive integer n, how many polynomials are there of degree n over Zz? How many distinct polynomial functions from Zz to Z2 are there? 9. Prove Corollary 2 of Theorem 16.2. 10. Let R be a commutative ring. Show that R[x] has a subring isomor- phic to R. 11. If q: R - S is a ring homomorphism, define d: R[x] - S[x] by (an x + . . . + do) - p(a,)x" + . . . + p(do). Show that o is a ring homomorphism. (This exercise is referred to in Chapter 33.) 12. If the rings R and S are isomorphic, show that R[x] and S[x] are isomorphic. 13. Let f(x) = 5x4 + 3x3 + 1 and g(x) = 3x2 + 2x + 1 in Zy[x]. Determine the quotient and remainder upon dividing f(x) by g(x). 14. Let f(x) and g(x) be cubic polynomials with integer coefficients such that f(a) = g(a) for four integer values of a. Prove that f (x) = g(x). Generalize. 15. Show that the polynomial 2x + 1 in Z4[x] has a multiplicative in- verse in ZA[x]. 16. Are there any nonconstant polynomials in Z[x] that have multi- plicative inverses? Explain your answer. 17. Let p be a prime. Are there any nonconstant polynomials in Zp[x] that have multiplicative inverses? Explain your answer. 18. Show that Corollary 3 of Theorem 16.2 is false for any commuta- tive ring that has a zero divisor. 19. (Degree Rule) Let D be an integral domain and f(x), g(x) E D[x]. Prove that deg (f(x) . g(x)) = deg f(x) + deg g(x). Show, by ex- ample, that for commutative ring R it is possible that deg f(x)g(x)