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# Basic Algebra

### Galois Theory / Solvability by Radicals

Show that 2x^5-10x+5 is irreducible over Q using Eisenstein's Criteria and show it is not solvable by radicals using typical results/theorems in Galois Theory/Solvability of Radicals in Galois Theory.

### Functions and Relations

A football player attempts a field goal by kicking the football. The ball follows the path modelled by the equation h=-4.9t2(means t squared)+10t+3, where h is the height of the ball above the ground in metres, and t is the time sincethe ball was kicked in seconds. 1. Describe the path of the ball. 2. After how many sec

### Do people use algebra in their jobs or their lives?

1. Very rarely do people use algebra in their jobs or their lives. At most, people use arithmetic. If this is the case then why do you suppose it is that we study algebra in the first place?

### Method found in the proof below to find a primitive elements

Use the method found in the proof below to find a primitive element for the extension Q(i, 5^(1/4)) over Q. *Recall that the extension field F of K is called a simple extension if there exists an element In this case, u is called a primitive element. **Theorem: Let F be a finite extension of the field K. If F is separabl

### Galois Theory: Algebraic Extension of a Perfect Field

Show that any algebraic extension of a perfect field is perfect (using the below hint only). Hint: Let K be a perfect field and F an algebraic extension of K. If F is not perfect, then there is a polynomial f(x) an element of F[x] that has an irreducible factor p(x) with a repeated root u. Here u is algebraic over K; let g(x)

### Galois Group Example Problem

Show the Galois group of (x^2-2)(x^2+2) over Q (rationals) is isomorphic to Z_2xZ_2 (Direct product group of integers modulo 2). For your choice of your mapping theta, that is operation preserving, 1 to 1, and onto please show that is well defined.

### How many different types of this shirt are made?

A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?

### Solve each equation and check for extraneous solution.

Solve each equation and check for extraneous solution. # 41. ----------------- / 2x^2 - 1 = x # 42. ------------------------ / 2x^2 - 3x - 10 = x Solve the equation # 85 ----------------------- / x^2 + 5x = 6 # 91 ----------------------- /

### Group Theory - Abelian Group: Show that if every element of the group G is its own inverse, then G is abelian.

Modern Algebra Group Theory (XX) Relation between Cyclic Group and Abelian Group Show that if every element of the group G is its own inverse, then

### Algebra: Solve 10 equations

Page 313 Solve each equation. # 8 ( a + 6 )( a + 5 ) = 0 # 9 ( 2x + 5 )9 3x - 4 ) = 0 # 12. t^2 + 6t - 27 = 0 # 48 - 2x^2 - 2x + 24 = 0 # 49 11 Z^2 + ---z = -6 2 # 52 3x (2x + 1 ) = 18 Page 364 Solve each equation. Watch for ext

### If the group G has four elements, show it must be abelian.

Modern Algebra Group Theory (XVI) Abelian Group If the group G has four elements, show it must be abelian.

### Splitting Fields : Let F be an extension field of K of degree 2, then F is the splitting field over K for some polynomial.

Let F be an extension field of K of degree 2, then F is the splitting field over K for some polynomial.

### Matlab Hermite Interpolation

(See attached file for full problem description with equations) --- I need some guidance on a Matlab programming project. The project concerns Hermite polynomial interpolation. We are given values and and their derivatives and at and respectively, and we look for coefficients such that if

### Mathematical Reasoning - Phil and Phyllis are Siblings.

Phil and Phyllis are siblings. Phyllis has twice as many brothers as she has sisters. Phil has the same number of brothers as sisters. How many girls and how many boys are in the family?

### Rational expressions

(See attached file for full problem description with equations) --- Pg.283 Factor the GCF in each expression 59. 60. Pg. 306 Factor each polynomial completely, if polynomial is prime, say so 35. 40. Pg 338 Perform indicated operation 22.

1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Please

### Description of Abelian group

Modern Algebra Group Theory (VII) To prove that if G is an abelian group, then for all a,b belongs to G and all integers n, (a.b)^n=a^n.b^n.

### Polynomial From Finitely Many Points on the Polynomial

Alice: "I'm thinking of a polynomial f(x) with non-negative integer coefficients. Can you tell which one?" Bob: "Well, I need some information." Alice: "You can pick any real number r and I'll tell you f(r). Um....that is, I'll tell you finitely many digits of f(r) - but as many as you want." Bob: "Gee - just one value

### Sample problem set

--- 1. Consider the following market demand and asymmetric cost functions for the airplane production industry. Market Demand is: P=200- (qA + qB) Cost Function for Boeing: C(qB) = 40 qB Cost Function for Airbus: C(qA) = 30 qA a.) Assume that the two act according to the Cournot mo

### To determine whether the system described is a group. G = set of all integers, a.b = a + b

Modern Algebra Group Theory (IV) To determine whether the system described is a group. G = set of all integers, a.b = a + b

### To determine whether the system described is a group

Modern Algebra Group Theory (II) Determine whether the system described is a group. G = a_0, a_1, a_2, a_3,...... where a_i.a_j = a_(i + j)

### To prove that G is a cyclic group of order n

Modern Algebra Group Theory (I) G contains all symbols a^i, i = 0,1,2,......., n - 1 where we insist that a^0 = a^n = e, a^i.a^j = a^(i + j)

### Graphs and Functions : Asymptotes, X- and Y-Intercepts

1. graph f(x) = + 5x+ 4 be sure to label all the asymptotes and to list the domain the x and y- intercept 2. f(x) = +3, x - sketch the graph and use the graph to determine whether the function is one to one -if the function is one to one find a formula the inverse 3. In ch

### Quadratic Equation Application Word Problems : Minimizing Cost

If a manufacturer of lighting fixtures has a daily production cost of (x)=800-10x+0.25x^2 where c is the total cost in dollars and x is the number of units produced. - How many fixtures, x, should be produced each day to minimize the cost? - What would it cost, c(x) to produce that many fixtures?

### Linear algebra: orthanormal basis

(See attached file for full problem description) --- For the A-matrix: 5x1 + 9x2 + 2x3 = 24 9x1 + 4x2 + x3 = 25 2x1 + x2 + x3 = 11 construct an orthonormal basis with a1 and then a2 and then a3. Next, expand the given vector b in terms of those vectors. ---

### Field Extension/Transcendental

Let F be an extension field of K. If u is an element of F is transcendental over K, then show that every element of K(u) that is not in K in also transcendental over K. Hint for proof: Suppose y is an element of K(u). Then for some g(x), h(x) elements of K[x], we have y = g(u)/h(u). Assume that y is algebraic over K and think

### Several Problems

(See attached file for full problem description) --- 2. Page 237, problem 102 Increasing deposits. At the beginning of each year for 5 years, an investor invests in a mutual fund with an average annual return of r. The first year she invest \$10; the second year, she invest \$20; the third year; she invests \$30; the fo

### The Employee Credit Union at Directional State University is planning the allocation of funds for the coming year.

The Employee Credit Union at Directional State University is planning the allocation of funds for the coming year. ECU makes four types of loans and has three additional investment instruments. Each loan/investment has a corresponding risk and liquidity factor (on a scale of 0-100, with 100 being the most risky/liquid). The v

### Distance Between Ships

At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 17 knots and ship B is sailing north at 19 knots. How fast is the distance between the ships changing at 4 PM, in knots? (1 knot is a speed of 1 nautical mile per hour) - The final result is not around 25 (i.e. 25.XX is not correct). Please check

### Logarithms : Express as a product: log7 4&#8730;[y]

Express as a product: log7 4&#8730;[y]