(4) (10 points) Show that 3 is a prime element in Zg]. Find the irreducible Z8]. Specify the irreducible factors that appear in the factorization of 9t. ation of 9i in let Prime P-(3) Thus 3 divide s Nea) ar 3 divides NC) ud we can sahat divides N(O)x possibility t, Hat bof ,, hak rosidue omod 3 Henle 9ie3 and p.3 (s prime. 3, us Sethat we are left wl oor 1 , So the only if you work...
b) Show that (1+i) is a zero of F(x) = 2x^5 - 9x^4 +12x^3 -4x^2 - 8x +4 c) Find all of the ZEROES of F(x)
OD (2) Show that if m>gedlm,a)>l, then [a] is a zero-divisor' in 2/m2 An element [a] in 2/m2 is a zero-divisor if [a] * [O] and there is a [b]+[O] in 2/m2 such that [a][b]=[0]. (3) Which elements in Z/m2 have multiplicative inverses? Hint: If d=acdm.a), then abed (mod m), for some b€2
6. [Purpose: Apply a new definition.] Classify each element of the given rings as a unit, zero divisor, or neither. (For example, in Zo, the units are [Il and [5l; the zero divisors are [2), [3), [4]; and [0) is neither a unit nor a zero divisor.) (b) Z (c) Z (a) Z12
Lab м м 281) Groph ye 1+2 cos (4x). Show a.) Cosine curre b.) dasked midlice c.) X-values of Start and end it. & full cycle.
Problem 2. Evaluate each of the following integrals. a) In(4x) dx 1 4x+1 dx ex b) Jo c) S (5x-2)e 6x dx
4X Evaluate 4x-4x+2)(x-1)2 ax. Carefully.
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
if-1<2x-1<1 determine the values of a and b
2 4x +3
The polynomial 4x^4-4x^3-19x^2+14x-3 has four rational zeros. Find the zero that has a multiplicity two. a: -1/2 b: -3 c: 1 d: -1 e: 1/2