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Number Theory

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  = . (a) Verify that A and B are bases of the Z-module P3. (b) Compute the change of basis matrices PAB (from the

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^

Palindromic and Reciprocal Polynomials

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 polynomial of degree m, th

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

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

Algebraic Number Theory

Problem 1: Prove that there are no integers x, y, and z such that x^2 +y^2 + z^2 = 999 Problem 2: Show that square root of 2 cubed is an irrational number. Problem 3: For each of the following pairs a and b, use the division algorithm to find quotient q and remainder r. (a) b=189, a=17

Polynomials : Roots, Holes and Vertical Asymptotes

#1: f(x)=(1x^3-18x^2+101x-168)/(1x^3-2x^2-41x+42) Find: All Roots:? All Holes:? All Vertical Asymptotes:? The answer to the 1st question is NOT 8 or -8. #2: f(x)= (-2x^3+37x^2-222x+432)/(1x^3+3x^2-64x-192) I need the roots: (I have two of them (9/2) and (6) but I need the third)? I need to know the hol

Simplifying Polynomials and Collecting Like Terms

Section 4.3 Collect like terms. 4 4 52. 3a - 2a + 2a + a Collect like terms and then arrange in descending order. 3 3 4 66. -1 + 5x - 3 - 7x + x + 5 Classify the polynomial as a monomial, binomial, trinomial, or none o

Binary Representations and Prime Factors

1- For n belongs to N (set of natural numbers) let B(n) denote the number of digits used in the binary representation of n. For example B(1) = 1; B(2) = 2; B(3) = 2; B(4) = 3: Find a closed formula for B(n) for an arbitrary n belongs to N. 2: Prove that if gcd(a, b) = d then a/d and b/d are relatively prime. 3- Find

Theory of Numbers : Fibonacci Number

Suppose that F1 = 1, F2 = 1, F3 =1, F4 = 3, F5 = 5, and in general Fn = Fn-1 + Fn-2 for n ≥ 3 ( Fn is called the nth Fibonacci number.) Prove that F1 + F2 + F3 +...+ Fn = F(n + 2) - 1

Primitive numbers

Assume that n is odd and a is a primitive root mod n. Let b be an integer with b ≡ a(mod n) and gcd (b, 2n) =1. Show that b is a primitive root mod 2n.