### Discriminants and Galois Groups

Compute the discriminants and the Galois groups of the polynomials x3 + 27x − 4 x4 − 5

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Compute the discriminants and the Galois groups of the polynomials x3 + 27x − 4 x4 − 5

Describe all subgroups of S5 which are Galois groups of irreducible polynomials of degree 5.

Find an irreducible polynomial defining the extension Q(3^1/2, 5^1/2).

Let K be obtained as a field Q(alpha) where alpha is a root of P(x) = x3 −3. Find an irreducible polynomial which defines the splitting field of P(x).

NUMBER OF FARMS. The number N of farms in the United States has declined continually since 1950. In 1950 there were 5,647,800 farms, and in 1995 that number had decreased to 2,071,520. Assuming that the number of farms decreased according to the exponential model: A). Find the value of k, and write an exponential function

Please see the attached file for the fully formatted problem. Show the existence of an extension of Fq of order l for any prime l.

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For the equation 24x^2 + 68x + 28 = 0 a) Find the discriminant b) If the discriminant tells you that you can factor, do so. c) Solve the equation by completing the square(HINT: at some point during the process, you will come to a point which looks like (x+(17/12))^2 = (121/144) d) solve the equation by using the quadrati

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1. Prove the Quadratic Formula by completing the square on ax^2+bx+c=0 2. A picture that is 14 in by 20 in is to have an even border. If the area of the picture and the border is 352 in^2, how wide will the border be?

Please see the attached file for the fully formatted problems. Simplify the following expression: 1/[1 + 1/(1 + (1/x))]

Please see the attached file for the fully formatted problem. (y2 + y - 12)/(y3 +9y2 + 20y)/[(y2-9)/(y2+3y2)]

Please see the attached file for the fully formatted problems. -6/x-2 / 8/3x-6

6) Suppose we have a building with a floor shaped like an isosceles right triangle. The two sides adjacent to the right triangle have length 100 feet. Think of the right angle being at the origin, and other two corners at (100, 0) and (0, 100). The overhead crane is located at the origin and needs to travel to a point (X, Y), w

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A human being has about 2.5 X 10(to the 13th power) red blood cells in her bloodstream. There are about 2 white blood cells for every 1,000 red blood cells. How many white blood cells are in a human's bloodstream? Please show your answer in scientific and standard notation.

4-terms - NOT factored by grouping. Factor out greatest common factor. 2-terms together. Get down to 3 terms first. 4t(xt+yt)+4t(x+y)-24x-24y

At Very Long Hotel in Florida, there are n rooms located along a very long corridor and numbered consecutively from 1 to n. One night after a party, n people, who have been likewise numbered from 1 to n, arrived at this hotel and proceeded as follows: Guest 1 opened all the doors. Then Guest 2 closed every second door beginnin

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Let n be a natural number, and let f(x) = x^n for all x are members of R. 1) If n is even, then f is strictly increasing hence one-to-one, on [0,infinity) and f([0,infinity)) = [0,infinity). 2) If n is odd, then f is strictly increasing, hence one-to-one, on R and f(R) = R. "This needs to be a proof by induction, provin

Find a nontrivial central Z_2 extension of the group A_4, meaning an extension of the form: 1 --> Z_2 --> G --> A_4 --> 1 Also, is it unique? The trivial extension is just the direct product of Z_2 and A_4.

Please see the attached file for the fully formatted problems. 1. The value of e ln 1+ ln 2+ ln 3 is ? 2. If logN(25) - logN(81) = 2, then N is? 3. Solve for x: log5( x) + log5(2x +13) = log5(24) 4. Find the coefficient of the fourth term in the expansion of (2x +3y)^11

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Describe all nonisomorphic central extensions of Z_n by Z_2 x Z_2 meaning central group extensions of the following form 1 --> Z_n --> G --> Z_2 x Z_2 --> 1 Meaning, determine those nonisomorphic groups G that can be described by such an extension. Please also explain how you came up with the answer.

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Solve for p where a=p(1+r)^t

Please see the attached file for the fully formatted problems. Problem 3: This problem itself is directly creating perceptual curiosity and at the cognitive level it is against what is known in the mathematics. The problem is: 13+23+33+.... +n3 = (1+2+3+...+n) 2 Visual representation of problem: Sum of the cubes

You have three thousand bananas that you have to get to a destination 1000 miles away. You can only carry 1000 bananas at a time. You also must eat 1 banana per mile for energy. Assuming you design your trip as efficiently as possible, how many bananas will you have left when you arrive at the destination? (apparently someon

If we assign number values to letters in the following way: A = 26, B = 25, C = 24 and so on until Y = 2 and Z =1, spell a word such that the product of its letters is as close to a million as possible. Explain how you went about solving this problem.

As provided by E. Galua theory the general algebraic equations for a polynomial of fourth order ax^4 + bx^3 + cx^2 + dx + f=0 (*) is the maximum order type of algebraic equations the solution to which one can write down in radical expressions. Among all the equations of fourth