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# Number Theory

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### Polynomials and Scientific Notation

Need the attached six (6) problems solved, so that I can then solve other similar problems. Instructions are in the attached word document. Subtracting polynomials. Show all steps in arriving at the answer. a. (t^2 - 6t + 7) - (5t^2 - 3t - 2) Multiplying polynomials. Show all steps in arriving at the answer.

### If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c. That is, if R is a unique factorization domain and if a and b in R are relatively prime (i.e., (a,b) = 1), whenever a divides bc then a divides c.

If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c. That is, if R is a unique factorization domain and if a and b in R are relatively prime (i.e., (a,b) = 1), whenever a divides bc then a divides c.

### Evaluating Polynomials

R(u)=8u^2-1/u^2-u-6,R(2)

### Evaluating Polynomials

P(x)=x^2-2x-3, P(-2)

### Taylor Polynomials Examined

Let p(x) = 7- 3(x-4)+5(x-4)^2 - 2(x-4)^3 + 6(x-4)^4 be the fourth degree of polynomial for the function f about 4. Assume f has derivatives of all real orders 1. Find f(4) and f'''(4) 2. Write the second degree Taylor polynomial for f' about 4 and its approximate f'(4.3). 3. Write the fourth degree Taylor polynomial for

### Evaluating Polynomials

1. The promoters of a county fair estimate that t hours after the gates open at 9:00 AM visitors will be entering the fair at the rate of -4(t+2)^3+54(t+2)^2 people per hour. How many people will enter the fair between 10:00 AM and noon? 2. A manufacturer estimates marginal revenue to be R(q) = 100q^-1/2 dollars per unit wh

### Greatest Common Divisor of Polynomials

Let f(x) = x^4+2x^3−x^2−4x−2 and g(x) = x^4+x^3−x^2−2x−2. Find the greatest common divisor d(x) of f(x) and g(x) in Q[x]. Find polynomials a(x), b(x) in Q[x] such that d(x) = a(x)f(x) + b(x)g(x).

### Palindromic Polynomials with Z Module Coefficients

I need to do some research on the properties of palindromic polynomials with Z(n) coefficients. I would like information/explanation of polynomials with Z(n) coefficients. I would like to see examples of polynomials with Z(1), Z(2), Z(3), Z(4), Z(5) and in general Z(n) coefficients. Also, I would like to see some examples of

### Prove that L_n = L_(n - 1) + L_(n - 2)

The Lucas numbers L_n are defined by the equations L_1 = 1 and L_n = F_(n+1) + F_(n-1) for each n > or = 2 Prove that L_n = L_(n-1) + L_(n-2) (n > or = 3) See attached file for full problem description.

### Vector Spaces and Subspaces

2. Use Theorem 5.2.1 to determine which of the following are subspaces of R3. Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold. (a) If u and v are vectors in W, then u + v is in W. (b) If k is any scalar and u is any vector in W,

### GCD of Two Polynomials

Let f(x) = and g(x) = . Find the gcd(f(x), g(x))) in Z[x] and express it as a linear combination of f(x) and g(x). Please see the attached file for the fully formatted problems.

### Irreducible Polynomials and Isomorphic Fields

Let F=Z7 and let p(x)=x^3 - 2 and q(x)= x^3 + 2 in F[x]. Show that p(x) and q(x) are irreducible in F[x] and that the fields F[x]/p(x) and F[x]/q(x) are isomorphic. See the attached file.

### Z-Modules of Polynomials, Basis and Linear Combinations

Please see the attached file for the fully formatted problems. Let P3 = ( it is set of all polynomials with coefficients in Z that are at most of degree 3.) Let A = and B = where , that is &#61506; = . (a) Verify that A and B are bases of the Z-module P3. (b) Compute the change of basis matrices PAB (from the

### Greatest Common Divisor of Polynomials : Euclidean Algorithm

Let f(x) =.... and g(x) = .... a) Find the gcd(f(x),g(x)) in Z[x] and express it as a linear combination of f(x) and g(x). b) Find the gcd(f(x),g(x)) in R[x] and express it as a linear combination of f(x) and g(x). See attached file for full problem description.

### Lagrange Interpolating Polynomials and Kronecker Delta

We are given the Lagrange polynomial in the form: P_n(x)= y_0*L_0(x)+...+y_n*L_n(x) and y_i = f(x_i) and L_i(x)= ((x-x_0)***(x-x_n)) / ((x_i-x_0)***(x_i-x_n)). We must show that L_0(x) + ... + L_n(x) = 1 for all x and n=3. Later we are to generalize this for all n>0. But let's just focus on n=3 for now; I may be able t

### Sets: Venn Diagrams and Natural Numbers

1. Suppose you have a Venn diagram showing three sets. Call the sets A, B, C. How many regions of the Venn diagram correspond to elements that are part of set A? Why so many? 2. Why is the set of Natural numbers an infinite set, but the set of blades of grass outside your residence a finite set? Explain.

### Polynomials : Algebraic Division and Complex Roots

The polynomial: 4x^4 - 6x^3 + 4x^2 - 3x + 1 has the real roots at x = 1 and x = 1/2 and two complex conjugate roots. (a) by the process of algebraic division and then solving a quadratic equation, find the complex roots. (b) Write down all the factors of the 4th degree polynomial 4x^4 - 6x^3 + 4x^

### Irreducible Polynomials

Prove that X^2-Y^2-1 is irreducible in Q[X,Y].

### Palindromic and Reciprocal Polynomials

Please see the attached file for the fully formatted problems.

### Solving Palindromic and Reciprocal Polynomials

See the attached file. 5. - Show that the product of a polynomial and its reciprocal polynomial is a palindromic polynomial. Hint Consider the zeros. Definition of reciprocal polynomial of f(x) for the book Introduction to the Theory of Error-Correcting Codes, by Vera Pless, 3rd edition Page 58 and 59. If f(x) is a po

### Primitive Irreducible Polynomials

Please can you explain primitive irreducible polynomials and please give examples. Please see the attached file for the fully formatted problem.

### Palindromic Polynomials

Problem #4 A palindromic polynomial is such that for all . Now we use the result in problem #3 to find an irreducible palindromic polynomial of degree 6 over . First, we compute all reducible ones. Now we consider the palindromic . From problem #3(c), is irreducible. I am sorry but I do not understand the definit

### Irreducible Palindromic Polynomials of Even Degrees

Show that if a palindromic polynomial of degree n is irreducible over F, then n must be even. Hint Experiment with palindromic polynomials of odd degree Please, can you explain what does palindromic polynomials means? Give me examples palindromic polynomials with even and odd degree.

### Irreducible Polynomials, Powers and Primes

1.-Let p a prime and let , (this is a extension field) , where is an irreducible polynomial over . Show that if are elements of that satisfy . Note this show that the pth powers of the elements of are distinct, and therefore every element in is the pth power of a unique element in . Therefore every element in has a un

### Irreducible Polynomials and the Euclidean Algorithm

Find the polynomials that represent 1/x^3+x , x/x^3+x, x^2/x^3+x, and x^3/x^3+x modulo the irrreducible polynomial x^5+x^2+1 over F2 ( the field with two elements 0 and 1) Your answers should be polynomials over F2 with degrees at most four. (Can you explain in here why at most degree four) Note: Use the Eucliden algorith

### Solving Irreducible Polynomials

Prove that a polynomial f(x) of degree 2 or 3 over a field F is irreducible if and only if f(a) different of 0 for all a belongs F. Hint: Use the following theorem that a polynomial f(x) has x-a as a factor if and only if f(a)=0. Please can you explain this step by step. and Can you give me examples. Can you explain why

### Taylor Polynomials and Error Formulas

Find the third Taylor polynomial P(x) for the function f(x) = (x ? 1) In x about X0 = 1. a. Use P1(O.5) to approximate f(0.5). Find an upper bound for error |f(0.5) ? P3(0.5)| using the error formula, and compare it to the actual error. b. Find a bound for the error |f(x) ? P3(x)I in using P3(x) to approximate f(x) on the inte

### Applications of Mean Value Theorem : Real and Simple Roots of Polynomials

Show that if the roots of the polynomial p are all real, then the roots of p' are all real. If, in addition, the roots of p are all simple, then the roots of p' are all simple.

### Business Math - Mathematics of Finance. Discussion of federal adjustments of primer interest rate.

(1) You are having a meeting with the employees of Financial Outsourcing, Inc. During the meeting, you were asked several questions. Give your opinion to the questions. - Why do you think the Federal government adjusts (raises or lowers) the prime interest rate? - What do you think are some of the effects of the adjustmen

### Polynomials

See attached file for full problem description.