### Palindromic and Reciprocal Polynomials

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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

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

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

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.

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

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

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

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

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.

(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

Please help with the following: Express 36 as a product of prime numbers.

Prime number factorization Express 45 as a product of prime numbers

When two numbers are prime numbers the least common multiple is a) 1 b) product of the two numbers c) one of the two numbers d) inf

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

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#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

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

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

Prove that (Fn+1)^2 - Fn Fn+2 = (- 1)^n

Fill in the function table. when f(x) = x^2 , f(x) = 2x - 1 and f(x) = x^2 - 2x + 1 x f(x) -3 -2 -1 2 4

Theory of Numbers (IX) Principle of Mathematical Induction Fibonacci Number Prove that

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

Meaning is given to the sum and product of all the natural numbers and then it is shown that: 1+2+3+4+... = -1/12 1*2*3*4*... = sqrt(2*pi)

Prove that 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2

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.

Show that for every positive integer n, 8n+1 is not prime.

Prove that there are no integers x, y, and z such that x^2 + y^2 + z^2 = 999.

See attached file. This problem is from the Text Book Mathematical Methods in the Physical sciences 2nd Edition by Mary. L. Boas. Chapter 12. Section 5 problem number 6

List all irreducible polynomial of degree 2, 3 and 4 over F22juu2332. Prove your assertion. Please see the attached file for the fully formatted problems.