Abstract Algebra Answers

3x+2y=13
2x-y=4

prove first isomorphism theorem

Suppose 2a + 5b = 10 for some nonnegative real numbers a and b. The geometric mean of a and b attains its maximum when b is equal to which number?

cos a/1+sina + 1-sina/cosa

Let alpha = (a1, a2, a3,...,ae) be a cycle of length L. Prove that alpha^2 is a cycle if and only if L is odd.

Let G be a group. Let H be a finite subset of G and let H be closed with respect to multiplication. Prove that H is a subgroup of G.

Let G = {(a, b) | a, b are real number, b != 0}. Define (a, b) * (c, d) = (a + bc, bd) for all (a, b), (c, d) belongs to G. Then (G, *) is a

a)Commutative group
b)non-commutative group
c)not a group
d)cyclic group.

A partial order <= is defined on the set S = {x,a1,a2,...,an,y} as x<=ai for all i and ai<=y for all i, where n>=1. Number of total orders on the set S which contain partial order <= is
a) 1
b) n
c) n+2
d) n!

520 is 100% so what % is 354?

Let Z[x] be the domain of all polynomials with integer coefficients. Consider the constant polynomial 2 and the polynomial x. Let S = {a(x)*2 + b(x)*x: a(x),b(x) w/in Z[x]}. (note: Z is integers symbol)
a)Prove S = {f(x) w/in Z[x]: f(0) w/in 2Z}
b)Prove S is an ideal in Z[x]
c)Prove there is no polynomial d(x) w/in Z[x] such thats S = {q(x)d(x): q(x) w/in Z[x]}. [This Z[x] is not a principle ideal domain]