### Use of the generating function for Legendre polynomials

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

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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.

Let P(z) and Q(z) be complex polynomials of degree m and n respectively, such that Q(z) has distinct roots z1...zn. Assume that n >= m+2 . Prove that P(z1)/Q'(z1) +...+ P(zn)/Q'(zn) (hint: evaluate ∫c P(z)/Q(z) dz for a suitable curve C).

1) How do I show x+1 is primitive? 2) How do I prove x^4+x^3+1 is an irreducible polynomial of degree 4 over Z mod?

17. The first three Legendre polynomials are P0(x) =1 , P1(x) = x and P2(x)=1/2(3x^2-1). If x = cos θ, then P0(cos θ)=1 and P1(cos θ)= cos θ. Show that p2(cos θ)= 1/4(3 cos 2θ +1). Book:- Differential Equations, by Dennis G Zill, page ,number 17.

1. The figure shows the graphs of f(x) = X 3 and g(x) =AX 3.What can you conclude about the value of a? 2. If f(x)= x(x+3)(x-1), use interval notation to give all values of x where f(x)>0. 3. If f(x) =x(x-1)(x-4)2 , use interval notation to give all values of x where f(x)>0. 4. Find the quotient and remainder of f(x) =

I had to prove 4 theorems two of them dealt with Abelian elements, automorphism of R2 under compontentwise addition. I want to keep my original work in tack as possble BUT I would like the follwing corrections made based on the following comments. THESE ARE THE ISSUES THAT NEED TO BE ADDRESSED The paper needs to provide

Bernoulli Polynomials are defined and then various properties are demonstrated. Key words: 1) Contour Integral Definition 2) Euler Numbers 3) Fourier Series 4) Evaluation of Riemann Zeta function for even integers 5) Stirling Numbers of the Second Kind

Chapter 3.1 Derivatives of Polynomials and Exponential Functions. I don’t understand how to get the answers provided. Please explain this step by step. Differentiate each function. Y = X2 + 4X +3 V(t)=t2 – 1 Z= A + BeY √ (X) 4√(t3) Y10 Answer Y’=3

A geologist you spoke with is concerned about the rate of land erosion around the base of a dam. Another geologist is studying the magma activity within the earth in an area of New Zealand known for its volcanic activity. One of the shortcuts they apply when doing calculations in the field is to use synthetic division. After t

(See attached file for full problem description) The problem is solved by integration and evaluation between the limits and algebraic manipulation

There are problems to test the commutativity, distributive property, definition of natural numbers. Please find the attachment for the problems.

Discuss an element of drinking water where you see a public health gap or need that should be addressed. You may bring up a specific case relevant to your community if you have one, or bring up a global issue.

Show that there are infinitely many algebraic numbers such that . If and are distinct algebraic numbers with = , then what are the possible values of ? Illustrate each value in your list by giving a specific choice of and . Please see the attached file for the fully formatted problems.

1. Let p be a poistive prime integer. Show that there is no rational number r such that p = r^2. Please see the attached file for the fully formatted problems. ---

Need help in proving the following. May need above assertion to prove. (See attached file for full problem description)

The number n = 15744539 is a product of two prime numbers. Find these two prime numbers if it is also given that φ(n) = 15736560. You may only use elementary functions on your calculator (adding, subtracting, dividing, multiplying, taking the square root). Show your work.

Consider the function f(x)=x/(ex-1)+x/2. (a) f has a so-called "removeable singularity" at x=0, where it is (so far) undefined. What value should we assign to f(0) to make f continuous at x=0? (b) With this taken care of, f actually has a Taylor series about x=0. Find the first 10 terms or so of this Taylor series (use C

Suppose that p is a prime number ≥ 3. We can write 1 + 1/2 + 1/3 + 1/(p-1) = a/b with (a,b) =1. Prove that p divides a. Please see the attached file for the fully formatted problem.

Suppose that p is a prime number. Show that 0, (p-1)!/2, (p-1)!/3, ..., (p-1)!/(p-1) is a complete residue system modulo p.

Suppose that p is a prime number ≥ 3 and r1, r2...rp is a complete residue system of modulo p. Prove that r1+r2+...+rp is divisible by p.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)

Problem 6. Suppose that f(s) = adxd+ad-1xd... is a polynomial with integral coefficients (so a0, a1 . . ,ad E Z). Show that f(n)?f(m) is divisible by n ? m for all distinct integers n and m.

Find Prime factorization (by hand) of 2006 and 2007.

2. (Taylor polynomials) (a) Write down the Taylor polynomials Pn(x) of degree n = 0, 1, 2,3 for the function f(x) = ln x about the point x = 1. (h) Plot the polynomials Pn(x) and the function f(x) on the interval [0, 3] using Matlab. Describe how the error |f(x) ? Pn(x)| behaves with respect to the point x and the degree n.

List 4 problems dealt with by cryptography & give real world examples of each. 2 paragraphs please.

(See attached file for full problem description) --- Suppose that a(x) and m(x) are relatively prime polynomials in F[x]. Bézout's Theorem guarantees the existence of polynomials u(x) and v(x) in F[x] such that... --- (See attached file for full problem description)

6301 is prime. If x, y, and z are integers that are not divisible by 6301, which of the following is equal to x^6299.y^12600.z^18903 mod 6301 ? (a) xyz (b) yz2/x2 (c) z3/x (d) 1/(x2 y2) (e) none of the above

Which of the following is true. a) 16 is a non-trivial square root of 1 modulo 51; hence 51 is composite b) 7 is a non-trivial square root of 1 modulo 47; hence 47 is composite c) 8 is a non-trivial square root of 1 modulo 55; hence 55 is composite d) all of the above e) None of the above

Q.1 For a number x, with 1< x < p, the number x^n mod p can be computed with at most 2log2 n modulo p multiplications. Asymptotic notation questions Q.2 2^(2n) = O(2^n) Q.3 log*n = O(log*(log n)) Q.4 The sqrt n th Fibonacci number can be computed and written in O(log n) time Please see the attac