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# Basic Algebra

### Algebraic Synthetic Division

See the attached file. In problems 1 through 4, write out the polynomial that has the listed factors: Example: (x), (x-1) ANS: x2-x 1. (x-2), (x+3) 2. x, (x-2), (x-1) 3. (x-2i), (x+2i) 4. x, (x-1), (x+1), (x-2) In problems 5 through 8, write the polynominal having the listed roots: 5. i 6. 2, 1,

### Simplifying exponential equations

32. Zero exponent. Simplify each expression. 5y^2z(y^-3z^-1) 38. Changing the sign of an exponent. Write each expression without negative exponents and simplify. 5^-2xy^-3 3x^-2 52. The quotient rule for exponents. Simplify each expression. 2r^-3t^-1 10r^5t^2∙t^-3 64. Use the rules of exponents to simplify eac

### measure theory: show is countable

Let {x_alpha}_[(alpha)(E)(gamma)] be an indexed collection of non-negative real numbers. The sum of this collection is defined to be the supremum of the set of all sums over finite subsets of gamma. This is The sum of_[(alpha)(E)(gamma)]x_(alpha) = sup{(the sum of)_(alpha)(E)(S)x_(alpha): S is finite subset of gamma} Prove t

### Miscellaneous algebra questions

1. Rationalize the denominator. √13 - √14 / √13 + √14 2. The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as its length. If a beam 1/4 foot wide, 1/2 foot high, and 16 feet long can support 30 tons, find how much a similar beam can sup

### Payoff and Opportunity Loss Tables

In the following payoff and opportunity loss tables circle and label the payoff or loss associated with the following decision-making criteria under uncertainty. a. maximax b. maximin c. equally likely d. minimax Payoff Table State of Nature Alternatives A

### Inequality in the criminal justice system concerning race

A discussion on inequality in the criminal justice system concerning race. Analyzed are the three videos listed below. Links: - http://www.nytimes.com/video/2012/06/12/opinion/100000001601732/the-scars-of-stop-and-frisk.html - http://abcnews.go.com/video/playerIndex?id=6953414 - http://www.youtube.com/watch?v=VQHdbW36XjE

### Divisibility property in mathematics

One of the most often misunderstood concepts in math is divisibility. Divisibility is a different operation than division and the two are often confused. The solution describes the seven key properties of divisibility and proves them mathematically to show the reader that they are true and why they are important. The seven

### Mathematics: Fundamental Algebra and Graphing

1. The cost, c, in dollars of a car rental is 10 + , where m is the number of miles driven. Graph the equation and use the graph to estimate the cost of car rental if the number of miles driven is 34. A) About 24 dollars B) About 15 dollars C) About 36.5 dollars D) About 19 dollars 2. Graph the two lines x + 2y

### Mathematical Modeling

Can you please explain the concept of modeling and how does a model describe known data and predict future data? Can you give me a specific example?

### Explanation of Mathematical Expressions Using Examples

I am confused with the concept of mathematical expressions. How can I tell when a collection of operations and numbers is an expression? Is "234x + - ÷ 14 - 23 46 -" an expression? Can you help me understand this concept? Thanks

### Applying Algebraic Formulas

1. Against the wind small plane 858 miles in 2 hours an 10 min. Return trip took 1 hr. and 50 min. What was the speed of the wind? What was the speed of the plane in still air? 2. A basketball scored 12 times during 1 game. He scored a total of 19 points, 2 for each 2 point shot, and 1 for each free throw. How many 2 pt.

### Induction steps for the following two algebra problems.

Hello, Can you please assist with the induction step for this problem? 3. Prove that 1 + 2 + 2^2 + ... + 2^(n-1) = 2^n - 1 for every n > 1. 5. Prove that for any real number x and for all numbers n > 1, x^n - 1 = (x - 1)(x^(n-1) + x^(n-2) + ... + x^(n-r) + ... + x + 1).

### Antibiotics Study

The following paragraphs are taken from the article, "Antibiotics' Role in Heart Attacks to Be Focus of Study." (Source: The Morning Call, April 5, 1999.) "Could taking an antibiotic spare heart patients future heart attacks, bypass surgery or death? "Researchers hope to find out with a large, federally funded study based

### Energy of a particle attached to 3 springs

I am having trouble with the type of question listed below and any help would be greatly appreciated. I have attached a diagram also. A block P of mass m is attached to three springs whose other ends are attached to fixed points A, B and C. I have listed the stiffnesses of the three springs and their natural lengths below. Th

### Complete Induction Proof Help

I'm currently working on this problem and I know that P(1) = 0, P(2) = 1, P(3) = 3, P(4) = 6 so P(i+1) = P(i) + i but I'm kind of confused how to structure it... Here is the complete question: Consider a 1-player game using a bag of n marbles. The player starts by dividing the bag of marbles into two groups (so that each g

### Simple Algebraic Relationship Between FKM(t) and FNA(t)

Using the approximation e^x=1+x for small x, find a simple algebraic relationship between FKM(t) and FNA(t) . Comment briefly on the relationship you have found.

### Function Multiplication, Equation of a Circle & Tangent Line

Please see the attached file for the complete solution. (1) The table gives the values of the functions f and g. Use the table to evaluate the expressions below. If there is not enough information given, state the information you would need to evaluate the expression. x 0 1 2 3 4 5 3 5 0 2 1 4 2 7 1 5 3 0 a.] g(f

### Zero Factor Property and Inequalities

The zero factor property: A) p^2 - p = 42 B) 16x - x^3 = 0 C) (x + 2)(x + 3) = 20 Solve each equation for y. Assume a and b are positive numbers: D) y^2 + ay + by + ab = 0 Applications: E) Tennis court dimensions. In singles competition, each player plays on a rectangular area of 117 square yards.

### Algebra Correction

Given (x+3)(x-2)=12 Jerry did the following: X+3=12 X-2=12 X=9 X=14 x={9,14} Why is Jerry's solution incorrect and how would you explain his error to him so that he could redo the problem correctly? [Use words and math sentences to give a clear explanation] You must explain: 1. The flaw in his reasoning.

### Every subsequence has a subsequence that converges to x

Let x be an element of the set of extended real numbers, and prove that if a sequence of extended real numbers is such that each of its subsequences has a subsequence that converges to x, then that sequence (itself) converges to x.

### Solving Equations and Solving for a Variable

Solving equations: Solve the linear expession, show your work and check your answer #26. 5x + 7 = 0 #32. 14 = -5x - 21 Solving a variable: Solve each formula for the specified variable #18. d = rt for r The language of functions: For each formula express y as a function of x. #22. y - x = 6 Solve each compound ineq

### Factoring Polynomial Problems

Factor each polynomial completely 1. 2a^4 - 32 2. 2m^3 - 250n^3 Factor each polynomial 3. 2y^2 - 17y + 21 Factoring by substitution 4. x^13 - 6x^7 + 9x Factor each polynomial completely. The variables used in exponents represent positive integers. #66 -4b^7 + 4b^4 + 3b How do you work out each problem, I

### Abstract Algebra: Homomorphisms, Isomorphisms, and Automorphisms

Problem 1. Prove that Z / <n> ≈ Z_n , where n ∈ Z and n > 1. Problem 2. Prove that θ : g --> a^{-1} ga for a fixed a ∈ G and all g ∈ G defines an automorphism of G. Problem 3. Prove if H is the only subgroup of order n in a group G, then H is a normal subgroup of G. ** Please see the attachment for formatted q

### Abstract Algebra: Prove Some Results About Subgroups

1. Define (C_G)(H) = {g is a number in G: g h = h g for all h is a number in H), where H is a subgroup of the group G. Prove that (C_G)(H) is a subgroup of G. Note: (C_G)(H) is called the centralizer of H in G. 2. Define (N_G)(H) = {g is a number in G: gH = Hg], where H is a subgroup of the Group G. Prove that (N_G)(H) is a s

### Abstract Algebra: Identity Element of the Group

1. Prove that is a is a number in G, a group, and ab = b for some b of G, then a = e, the identity element of the group. 2. Consider the set of polynomials with real coefficients. Define two elements of this set to be related if their derivatives are equal. Prove that this defines an equivalence relation. 3. Let H be a s

### Standard Form and Factored Form

Please refer to the attached document for the full problem set. 20. Consider the polynomial P(x), shown in both standard form and factored form. P(x) = (1/10)x^4 - (1/2)x^2 + (41/10)x - 3 = (1/10)(x+3)(x-1)(x-2)(x-5) (a) Which sketch illustrates the end of behaviour of the polynomial function? (b) State the y-intercept.

### Exponential generating function for the number of (ordered) partitions of {1,...,n}

An ordered partition of [n]={1,...,n} is a partition (B_1,...B_k), where the order of the blocks matter. (Thus ({1,2},{3}) and ({3},{1,2}) are different ordered partitions of .) Let OS(n,3) be the numbered partitions of [n] into 3 nonempty blocks. Thus OS(n,3)=3! S(n,3). a) Find an explicit formula for the exponential gene

### Comparing Coefficients of Different Scales

Definition Let me define the influence of an independent variable on the dependent variable as the change in the dependent variable due to the independent variable. Background Suppose that your company has a nationwide hiring program, and you have determined with data consisting of SALES (\$M) in the first year on the job, u

### Binomial Coefficients - Positive Integer

Abstracts should not be duplicates of the question. Show that if n is a positive integer, then [see the attached file for the equation) a) using a combinatorial argument. b) by algebraic manipulation.

### Math: Kinetics Questions

Questions: 1. A hockey puck is hit on a frozen lake and starts moving with a speed of 13.1 m/s. Five seconds later, its speed is 6.9 m/s. What is its average acceleration? The acceleration of gravity is 9.8 m/s. Answer in units of m/s/s. 2. What is the average value of the coefficient of kinetic friction between puck and ice