Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is an integer.

a. tan ^ -1(y/x) Show that the function u(x,y)define classical solution to the 2-dimentional Laplace equation Uxx+Uyy =0

b. e ^ -(x-2t)^2 Show that the function u(t,x) is a  solution to wave equation

What is the order and type of the ff PDE's.

1. (1+X^2)U_xxyy-2xy^3U_xyyy+(1+u^2)U_YY=0

2.(1+U^2)U_xx-2U^2U_xy+(1+u^2)U_YY=0

3.(1+X^2_Y)U_xx-2U_XUyUxy+(1+u^2_X)U_YY=0

The least common multiple of nonzero integers a and b is the smallest positive integer m such that a|m and b|m. It is denoted [a, b], or sometimes [a, b] for short. Prove the following:

1) If a|k and b|k, then [a, b]|k.

2) If gcd(a, b) = 1, then [a, b] =ab

3) If c >0,then [ca, cb] =c·[a, b].

4) If a >0 and b >0,then [a, b] =ab / gcd(a, b).

Discuss modern mathematical theories such as dynamical systems theory, chaos theory, hyperbolic geometry, fractal geometry, spherical geometry.

In about 250 words discuss the history of foundations of mathematics as proposed in the late 19^th and early 20^th centuries.

Discuss the ideas of any three of the following:

1. Galois
2. Bolzano
3. Weierstrass
4. Leibniz
5. Lambert
6. Boyle
7. Hilbert
8. Brouwer
9. Feynman
10. Gödel
11. Touring

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring of the functions from X to R. Show directly the associativity of the multiplication of F(X, R). Assume that R is unital and commutative. show that F(X, R) is also unital and commutative.

Consider the ODE u" + lambda u=0 with the boundary conditions u'(0)=u'(M)=0, where M is a fixed positive constant. So u=0 is a solution for every lambda,

Determine the eigen values of the differential operators: that is

a: find all lambda such that the above ODE with boundary conditions has non trivial sol.

b. And, what are the non trivial eigenvalues you obtain for each eigenvalue

Which one of the following rectangular regions guarantees the uniqueness of a solution to the I. V. P. (10 x-13y) y′= 2x, y(26)=0.

Frank, Sofia, Eldridge, and Jake are the four qualifiers for a charity raffle with two $500 prizes. One of their names will be drawn for the first prize then replaced, at which point the second prize winner will be drawn. Draw a tree diagram to determine the sample space and find the probability that (a) One person wins both prizes. (b) There are two different winners. (c) Sofia wins at least one prize. (d) Frank wins both prizes. (e) The two winners are Jake and Eldridge.

6. For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to find matchings in graphs in general.

1. For which n does the complete graph Kn have a matching?
2. Prove that if a graph has a matching, then |V||V| is even.
3. Is the converse true? That is, do all graphs with |V||V| even have a matching?
4. What if we also require the matching condition? Prove or disprove: If a graph with an even number of vertices satisfies |N(S)|≥|S||N(S)|≥|S| for all S⊆V,S⊆V,then the graph has a matching.

Please keep straight to the point and short if possible, I give good ratings on good legible writings and correctness. THANKS!!

1. Problem 10-07 (Algorithmic)

   Aggie Power Generation supplies electrical power to residential customers for many U.S. cities. Its main power generation plants are located in Los Angeles, Tulsa, and Seattle. The following table shows Aggie Power Generation's major residential markets, the annual demand in each market (in megawatts or MWs), and the cost to supply electricity to each market from each power generation plant (prices are in $/MW).

   Distribution Costs
   City	Los Angeles	Tulsa	Seattle	Demand (MWs)
   Seattle	$364.25	$601.75	$67.38	958.00
   Portland	$367.25	$604.75	$189.13	842.25
   San Francisco	$166.13	$463.00	$284.88	2363.00
   Boise	$341.25	$460.00	$281.88	578.75
   Reno	$241.50	$479.00	$360.25	954.00
   Bozeman	$428.63	$428.63	$309.88	506.15
   Laramie	$367.25	$426.63	$367.25	1198.50
   Park City	$375.25	$375.25	$494.00	622.25
   Flagstaff	$238.13	$535.00	$653.75	1178.19
   Durango	$363.25	$303.88	$600.75	1472.25

   1. If there are no restrictions on the amount of power that can be supplied by any of the power plants, what is the optimal solution to this problem? Which cities should be supplied by which power plants? What is the total annual power distribution cost for this solution? If required, round your answers to two decimal places.
      
      The optimal solution is to produce  MWs in Los Angeles,  MWs in Tulsa, and  MWs in Seattle. The total distribution cost of this solution is $  .
      
   2. If at most 4000 MWs of power can be supplied by any one of the power plants, what is the optimal solution? What is the annual increase in power distribution cost that results from adding these constraints to the original formulation? If required, round your answers to two decimal places.
      
      The optimal solution is to produce  MWs in Los Angeles,  MWs in Tulsa, and  MWs in Seattle. The total distribution cost of this solution is $  . The increase in cost associated with the additional constraints is $  .

Consider the system of equations

2x-5y=a

3x+4y=b

2x- 4y=c

where a, b, c are constants. Because there are 3 equations and 3 unknowns, there are no possible values of a, b and c for which the system of equations has a unique solution. True or false?