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Number Theory

Evalute the polynomial

P dollars is invested at annual interest rate r for 1 year. If the interest is compounded semiannually, then the polynomial P(1 + r/2)2 represents the value of the investment after 1 year. Rewrite this expression without parentheses. Evaluate the polynomial if P = $200 and r = 10%.

Proof Techniques: Proof with Counterexample

Proof Techniques: Homework 03 Provide counterexamples to each of the following. Every odd number is prime. Every prime number is odd. For every real number x, we have x2 > 0. For every real number x  0, we have 1/x > 0. Every function f :ℝℝ is linear (of the form mx + b).

Calculation of LCM

The Week One Discussion will concentrate on the mathematical fact that all numbers in our real number system are the product of prime numbers. This fact alone is amazing. a. You will select the ages of two people in your life, one older and one younger. It would be great if the younger person was 15 years old or less.

Solving Polynomials and degree of polynomials.

From the given polynomials, identify the polynomials of degree one. a. 11y2 - 5 - 4y b. (3x2)1/2 + 12 c. 7 - (12)1/2x d. 2x + 13x2 e. 5x + 7y + 8 f. (12)1x1 g. x3 + 2x - 10 h. 3x + 4x - 4 Solve the following: i. 2x = -3x + 9 ii. 3x/5 = -6 iii. y/4 + 2 = 7 iv. 16 = -2x/3 v. Find f(1) for f(x) = 2x3 - 3x2

Classifying polynomials as monomials, binomials and trinomials

After completing please Classify the 15 given polynomials as monomials, binomials, trinomials, and polynomials. Use the format given below for categorizing the polynomials. Remember to simplify BEFORE you classify! - a. 2x3+3x2 + 2x - 1 b. ax4 - bx2 c. 3a4 - 6a3 + 3a2 d. -4xyz e. 3ab - 7cd + 5ac f.

Factoring Expressions and Solving Equations

Please see the attached file for the fully formatted problems. 1. Factor completely. 25x2 + 80x + 64 A) Prime B) (5x + 8)2 C) (5x + 8)(5x - 8) D) (5x - 8)2 2. Factor completely. 15z2 + 14z - 8 A) (15z + 4)(z - 2) B) (3z - 4)(5z + 2) C) Prime D) (3z + 4)(5z - 2) 3. Write the expression in lowest t

Polynomials and solving slopes

1. What is the coefficient of x^5 in the polynomial -8x^5+0x^3+14? 2. solve -43<20-9v<=-7 3. complete ordered pair to satisfy the given equation. 2x-5y= -12:(-1, __) 4. Find the slope of the line that contains points (-4,-2) and (1,1)

Number Theory : Quadratic Residues

Please provide a detailed solution to the attached problem. Please do not give a trivial answer. I think the questions asks us to determine the what form p is of (for example, p is a prime of the form 3n+1 (this was an example randomly chosen). I am trying to solve this problem myself (using that x^2-6 = 0 (mod p) => 6 is a quad

Solving Polynomials Applications

1. Three consecutive even integers are such that the square of the third is 76 more than the square of the second. Find the three integers. 2. A rectangular parking lot is 50 ft longer than it is wide. Determine the dimensions of the parking lot if it measures 250 ft diagonally.

Fundamental operations on polynomials

Answer the following questions involving a. Find P(-1/2) if P(x) = x4 + 3x2 + 2 b. Simplify (x2 + y2 + 4z2 + 2xy + 4yz + 4zx) + (x2 + 4y2 + 4z2 - 4xy - 8yz + 4zx) c. Simplify (3x + 2y)^2 d. Find the product (x + 9) (x - 4)

Yielding a Composite Number

A formula that yields prime numbers. One such formula was x^2 - x + 41. Select some numbers for x, substitute them in the formula, and see if prime numbers occur. Try to find a number for x that when substituted in the formula yields a composite number.

C++ - polynomials implementation

Design and implement a class for dealing with polynomials. The polynomial a(n)x^n + a(n-1)x^(n-1) + . . . + a0 will be implemented as a linked list. Each node will contain and int value for the power of x and an int value for the corresponding coefficient. The polynomial operations should include addition, multiplication, an

Rational function and polynomials

The reason why polynomials are so important is that there is a theorem from Analysis that says that any continuous function defined on an interval of the real line can be approximated arbitrarily closely by a polynomial. So polynomials are useful to ¿model¿ any kind of function on a closed interval. However, polynomials ¿ge

Polynomials and rational function

Now consider a rational function, which is the ratio of two polynomials. These two polynomials will each have a set of zeros, and note that at a zero of the denominator we are actually dividing by zero. The zeros of the denominator are called poles, and they are point where the rational function becomes infinite (unless there

Exponents and Polynomials

Solve each problem. 78. Swimming space. The length of a rectangular swimming pool is 2x -1 meters, and the width is x +2 meters. Write a polynomial A(x) that represents the area. Find A(5). 86. Selling shirts. If a vendor charges p dollars each for rugby shirts, then he expects to sell 2000 - 100p shirts at a tournamen

Exponents and Polynomials

Simplify. All variables represent nonzero real numbers. 1.) -5y^4 (y^5)^2 15y^7 (y^2)^3 2.) 3y^8 This whole problem is ^4 2zy^2 3.) 3^3 This whole problem is ^4

Time Sequence diagrams

Please reference this figure for the question: 55. Draw 3 time sequence diagrams that illustrate the flow of frames between points B and G in Figure 1 using the following information: a. Stop and Wait ARQ, SENDER sends four frames (F1, F2, F3, and F4). F1, F2 and F4 are received error free, but F3 is lost in the

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

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^

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