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# Number Theory

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### Polynomials: Multiplication, Division, Subtraction, Addition

See attachment, thank you very much. Simplify each expression 1. 2x3 . 3x2 (x3 = X raised to the power 3, x2 = X raised to the power 2). 2. (2x2 y)4 3. 18x4 divided by 3x or 18x4 ----------- 3x 4. x-9 = 5. 9x

### Polynomials Factored Completely

Factor completely (see attached) Factor Completely a. b. c. d. (11x-6y) (11x + 6y ) Simplify a. b. c. d. Multiply a. b. c. d. Solve this system of equations. 7x - 5y = 1

### Written Numbers in Expanded Forms

Please assist with attached math problems. 1 Write this number in expanded form a. 247,089 b. What digit tells the number of thousands c. What digit tell the number of ten thousands 2 Write this number in words \$8,886 3 Name the property of addition and explain your choice. a. 3+(0 + 6) = (3 + 0) + 6 b. 0 + a

### Problems with a Set Theory in Real Analysis

Undergraduate senior level Real Analysis. Please show me formal math proofs. Show that Q(set of rational numbers) ~ N(set of natural numbers). (Suggestion from my professor: prove by letting set Qn < Q and a/b (a and b are some integers, but b is not zero) and Q = union of Qn (Qn = Q1, Q2, Q3, ....) .)

### A Set Theory in Real Analysis

Formal Math Proofs Prove that each of the following sets is countable: a) The set of all numbers with two distinct decimal expansions (like 0.500... and 0.4999...); b) The set of all rational points in the plane (i.e., points with rational coordinates); c) The set of all rational intervals (i.e., intervals with ratio

### Marriage penalty eliminated ...

Marriage penalty eliminated. The value of the expression 4220 + 0.25(x - 30,650)is the 2006 federal income tax for a single taxpayer with taxable income of x dollars, where x is over \$30,650 but not over \$74,200. a) Simplify the expression. b) Find the amount of tax for a single taxpayer with taxable income of \$40,000. c) W

### Regular bipartite graph

Consider the sets A0 := {0, 1, 4}, B0 := {0, 2, 8}. Consider the sets Ai := A0 + i := {i, i + 1, i + 4} ,and Bi := B0 + i := {i, i + 2, i + 8}, for i = 1, 2, . . . , 12. All addition here is performed modulo 13. Consider the bipartite graph G whose vertices are the sets A := {Ai : 0 <= i <= 12} and B := {Bi : 0 <= i <= 12} (so

### Advance Products, Inc. has just organized a new division to manufacture and sell specially designed tables using select hardwoods for personal computers.

Advance Products, Inc. has just organized a new division to manufacture and sell specially designed tables using select hardwoods for personal computers. The divisions monthly costs are shown the schedule below: Manufacturing costs: Variable costs per unit: Direct materials \$86 Variable manufacturing

### Polynomial long division

See file (14x^3 -43x^2 -56x -15) divided by (7x+3)

### Exercise on Polynomials

Solve the following questions involving fundamental operations on polynomials a. Find p(x) + 4q(x) p(x)=4x^4 + 55x^3 - 23x^2 + 13 q(x)=43x^4+ 14x^2 -12 b. Find P(-1/2) if P(x) = 2x^4 + x^3 + 12 c. Simplify: (-4 + x^2 + 2x^3) - (-6 - x + 3x^3) - (-6y^3 + y^2) d. Add: (2x^2 + 6y^2 + 4z^2 + 3xy + yz + zx) + (4x^2 + 3y^

### Mathematics - Algebraic Number Theory..

If A and B are subsets of a set X, define their symmetric difference by A+B = (A-B) union (B-A) (i). Prove that A + A= empty set (ii) Prove that A + empty set= A (iii) Prove that A + (B+C) = (A+ B) + C (iv) Prove that A intersection (B+C) = (A intersection B)+(A intersection C)

### Mathematics - Algebraic Number Theory - Pigeonhole Principle

1a. If X and Y are infinite sets with the same number of elements, show that the following conditions are equivelent for a function f: X-->Y: (i). f is injective (ii). f is bijective (iii) f is surjective 1b. Suppose there are 11 pigeons sitting in some pigeonhole. If there are only 10 pigeonholes prove that there is a h

### Prime factors 98

Write 98 as a product of prime factors

### 100 as a Product of its Prime Factors

Please help me write 100 as a product of prime factors. Thank you for the help!

### 84 as a Product of Prime Factors

Please write the number 84 as a product of its prime factors.

### Performing Division Calculations

Perform the following division: (9x^3 - 19x + 8)/(-3x +4). Write your answer in the form Q(x) + (R(x))/(-3x + 4), where Q(x) is the quotient and R(x) is the remainder of the division.

### Polynomial division

See attachment Perform the following division: (6x^3 - 13x^2 + 11x + 10) / (3x - 5) Write your answer in the form Q(x) + R(x)/(3x -5) , where Q(x) is the quotient and R(x) is the remainder of the division.

### Telephoto

Using formula 1/f = 1/o + 1/i Find the image distance i for an object that is 2,000,000 mm from a 250-mm telephoto lens

### Contemporary Abstract Algebra, Author: Joseph A. Gallian

Chapter 0 (Preliminaries) Q.) How would you prove the converse? A partition of a set S defines on equivalence relation on S. Hint: Define a relation as X - Y if X and Y are elements of the same subset of the partition. 10.) Let n be fixed positive integer greater than 1. If a mod n = a' and b mod n = b' .Prove that (a

### Exponents, Multiplication, Division of Polynomials

See attached. 1. Evaluate the expression. Assume 2. Evaluate when y = -2 3. Evaluate when 4. Express the following using a positive exponent. Then simplify the expression . Write using a positive exponent do not evaluate 5. Express using a positive exponent = 6. Multiply and simplify =

### An overview on tessellations.

An overview of the theory and results on tessellations of three types of Riemann surfaces: the Euclidean plane, the sphere and the hyperbolic plane. Roughly speaking, a tessellation of a space is a pattern which, repeated infinitely many times, fills the space without overlaps or gaps . From a mathematical point of view, tesse

### How a Calculator Can Be Used as a Tool for Learning

Do you think that there are ways to use the calculator to build student's understanding of important math concepts?

### Algebraic Number Theory

Explain why the polynomial H(x) = x^5 + x^4 + x^3 + x^2 + 1 is irreducible over the integers.

### Members of the Given Real Number Subset

Directions: List all numbers from the given set B that are members of the given Real Number subset. Please explain. B=[ 19, square root 8, -5, 0, 0.7 as a repeating decimal, square root of 9] Integers B= [ 17, square root of 5, -2, 0,0.7 as a repeating decimal, square root 16,] Whole numbers B= [ 6,square root v8, -1

### Prime Numbers, Factors, Division

1. List the first seven prime numbers of our standard number system (0, 1, 2, 4, 4, etc...) 2. List all the factors for the number 72 3. Which of the numbers 21, 88, 126 and 255 are divisible by 7.

### Multiplying or Squaring Polynomials

Please help me with the following question: Multiply (x + 1/3)^2.

### Practice problems on Polynomials and Rational Functions

1. Explain how synthetic division may be used to find the factors/zeros of a polynomial function. Give an example of how this is accomplished. Use synthetic division to find the function value. 1) f(x) = 2x4 + 4x3 + 2x2 + 3x + 8; find f(-2). Write the quadratic function in the form y = a(x - h)2 + k. 2) y = x2 - 2x - 9

### A power of a prime number

Any help would be appreciated. Please see the attached file.

### Modeling using Second and Third Order Polynomials

To explore the versatility of rational functions, choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function. Provide a graph for the second-order rational function (e.g., x2), choosing x values in the range from -10 through +10. Then, provide at least three variations of th