Explore BrainMass
Share

Number Theory

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.

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

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,

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

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.

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 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

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

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