(7) Let

x^(4)-16

be an element of the polynomial ring

Z[x]

and use the bar notation to denote passage\ to the quotient ring

Z(z)/(x^(4)-16)

.\ (a) Find a polynomial of degree

that is congruent to

9x^(13)-7x^(9)+4x^(5)-2x^(3)+3

modulo

(x^(4)-16)

.\ Hint: long division.\ (b) Prove that

/bar (x-2)

and

/bar (x+2)

are zero divisors in

Z(z)/(x^(4)-16)

.

7) Let x416 be an element of the polynomial ring Z[x] and use the bar notation to denote passage to the quotient ring Z[z]/(x416). (a) Find a polynomial of degree 3 that is congruent to 9x137x9+4x52x3+3 modulo (x416). Hint: long division. (b) Prove that x2 and x+2 are zero divisors in Z[z]/(x416)