### Primitve Polynomials

Please do prob 33(a), 33(b) and 35 attached. Please provide explanations.

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Please do prob 33(a), 33(b) and 35 attached. Please provide explanations.

(r^2 -5r+9) / (r-3)

5^1367 mod 50174 How would I solve this? Would I need to use square and multiply?

I need a detailed solution for this exercise. For this problem, "k" represents an arbitrary field. Please see attachment.

Solve the following questions involving fundamental operations on polynomials: A) find p(x)+4q(x) p(x)=3x^5+70x^3-67x^2+3 q(x)=3x^3+56x^2-19 B) Find P(-1/2)if P(x)=x^4=3x^2+2 C) If P(y)=10y^2+4 Q(y)=y^3-5y^2+3y+7 Find 3P(y)+Q(y) D) (x^2+y^2+4z^2+2xy+4yz+4zx)+(x^2+4y^2+4z^2-4xy-8yz+4z

1.Add the polynomials. (x^3+4x^2+2x-3) + (-x^3-3x^2-3x+2) 2.Perform the indicated operation. (2 - 3x + x^3) - (-1 - 4x + x^2) 3.Find the product . -2ab*7a^5*b^4 4.Find the opposite of the polynomial. -2r^2 - r + 3 5.Multiply: (-4 - a)^2 6.Find the quotient and the remain

1) (-2)(-2)x+¹ 2) -2(3x²y³)² 3) -2²(3xy²)³

Theory of Numbers (XIV) Principle of Mathematical Induction Fibonacci Number Lucas number The Lucas numbers L_n are defined by the equations L_1 = 1 and L_n = F_(

7, 28, 24, 45

Please see the attached file for the fully formatted problem.

Throughout this exercise, g(x) = cos x and Pn(x) (n is a subscript) is the Taylor (Maclaurin) polynomial of order n based at x = 0. a) Find formulas for Pο (ο is a subscript) through P6 (6 is a subscript), the Maclaurin polynomials through order 6 for g based at x = 0. Please label on graph b) Is g odd, even,

First, find the Taylor polynomial Pn (n is a subscript) of order n for the function f with base point Xο (ο is a subscript). Then plot both f and Pn on the same axes. Choose the plotting window to show clearly the relationship between f and Pn. f(X) = sin X + cos X, n = 4, Xο (ο is a subscript) = 0

The degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and + 4i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.

I am struggling to figure out what sign goes on the top and if I am supposed to add or subtract. Can anyone help? See attached file for full problem description.

6.5 A half adder is a combinational logic circuit that has two inputs, x and y, and two outputs, s and c, that are the sum and carry-out, respectively, resulting from the binary addition of x and y. (a) Design a half adder as a two-level AND-OR circuit. (b) Show how to implement a full adder, as shown in Figure 6.2a, by usin

In F[x] let V_n be the set of all polynomials of degree less than n. Using the natural operations for polynomials of addition and multiplication, V_n is a vector space over F. Any element of V_n is of the form a_0 + a_1x + a_2x^2 + ... + a_(n-1)x^(n-1) where a_i belongs to F. Let F be

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.

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).

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

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.

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,

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.

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

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^

Please see the attached file for the fully formatted problems.

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.