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# Number Theory

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### Dividing Polynomials

1) Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express P in the form P(x) = D(x) · Q(x) + R(x). P(x) = 3x^2 + 4x â?' 1, D(x) = x + 5 P(x) =(x+5)(_____)+______ 2) Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and

### Expectations theory for interest rates

Expectations theory: what should be the interest rate on 3-year, risk-free securities? The real risk-free rate of interest is 3 percent. Inflation is expected to be 2 percent this year, and 4 percent during the next two years. Assume that the maturity risk premium is zero. What is the yield on 2-year Treasury Securities? W

### Show that prime element R generates a prime ideal.

Let R be a unique factorization domain then show that prime element R generates a prime ideal.

### Number theory and the real number system.

Why does Ellie see these pulses as a sign of intelligent life or what is the significant about the number of pulses? Contact If you could send a long message to such extraterrestrial beings - words, pictures, sounds, music - what would you say? How would you describe us? What would you leave out? Could you communicate intell

### Polynomials: Forming a Group Under Addition: Example Question

Does the set of polynomials P(x,2) with complex coefficients form a group under addition? State your reasoning. Also, what is the inverse element of (2i)x^2 - (3 + 4i)x + 1? Thanks very much.

### 1. For each of the following statements, write the contrapositive statement, and prove the original statement by proving its contrapositive: (a) If m^2 + n^2 &#8800; 0, then m &#8800; 0 or n &#8800; 0. 2. What is wrong with the following proof of the conjecture "If n^2 is positive, then n is positive.": Proof: Suppose that n^2 is positive. Because the conditional statement "If n is positive, then n^2 is positive" is true, we can conclude that n is positive.

1. For each of the following statements, write the contrapositive statement, and prove the original statement by proving its contrapositive: (a) If m^2 + n^2 &#8800; 0, then m &#8800; 0 or n &#8800; 0. 2. What is wrong with the following proof of the conjecture "If n^2 is positive, then n is positive.": Proof: Sup

### Simple Math Calculations in Real-Life Scenarios

1. Solve the problem. If one book of stamps lasts a family three months. How many books of stamps would be needed to last the family a year? Show your step by step work! Include correct units with your solution. 2. Solve the problem. If one book of stamps lasts a family three months. How many books of stamps would be needed t

### operations on polynomials

Please perform operations on polynomials using addition, subtraction, multiplication, and division. I will need 3 examples of each.

### 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%.

### Winter wheat. While finding the amount of seed needed to plant his three square wheat fields, Hank observed that the side of one field was 1 kilometer longer than the side of the smallest field and that the side of the largest field was 3 kilometers longer than the side of the smallest field. If the total area of the three fields is 38 square kilometers, then what is the area of each field?

Winter wheat. While finding the amount of seed needed to plant his three square wheat fields, Hank observed that the side of one field was 1 kilometer longer than the side of the smallest field and that the side of the largest field was 3 kilometers longer than the side of the smallest field. If the total area of the three

### 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

### Like Terms and Dividing Polynomials

1. When simplifying like terms, how do you determine the like terms? Please show an example. 2. Explain how division of real numbers corresponds with division of polynomials. Please show an example.

### Cryptography, congruences, and primes

Hi, Can you help me with these questions? Consider the set of all even integers 2Z=....If this factorization into primes can be accomplished, is it unique? (see attached)

### 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

### Properties of Natural Numbers

Let m denote a natural number. Use any properties of the natural numbers to argue that m x m is not equal to m + m.

### Primality of Mersenne Numbers

Determine whether m_13 = 2^13 - 1 = 8191 and m_23 = 2^23 -1 = 8388607 are prime.