Abstract Algebra Answers

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]

There is 4 ml of grape flavoring to every 100ml of 200mg Amoxicillin. The patient wants apple. You can add 1/5 the amount of apple as you can grape. How many milliliters of apple would you use to flavor 300 ml of Amoxicillin?

Translate the phrase to an algebraic expression. State what each variable represents.
one-fifth of a number plus one-sixth of a different number