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Number Theory

Congruences

5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.

Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if λ1...λn are such real numbers that the equality ... holds true for all x E R then .... Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and co

Finding area & factoring

I am trying to figure a problem that involves the area of a deck. If a deck is rectangular and has an area of X2 + 6x + 8 (the 2 is squared) square feet and a width of x + 2 feet. I am trying to determine the lengths of the deck.

Solving Polynomial Functions

I hope you will be able to help me with this problem, I'm really stuck. 10-(k+5) = 3(k+2) I would also appreciate any explanation.

Explain the Proof Step-by-step : Irrational Function

Recall that a perfect sqaure is a natural number n such that n = (k^2), for some natural number k. Theorem. If the natural number n is not a perfect square, then n^(1/2) is irrational. Proof. S(1): Suppose n^(1/2) = r/s for some natural numbers r and s. S(2): We may assume that r and s have no prime factors in common,

Prove that there is a one-to-one correspondence between the power set of a countably infinite set A and the set S of all countably infinite sequences of 0's and 1's, and that the power set of A is an uncountable set.

For any set B, let P(B) denote the power set of B (the collection of all subsets of B): P(B) = {E: E is a subset of B} Let A be a countably infinite set (an infinite set which is countable), and do the following: (a) Prove that there is a one-to-one correspondence between P(A) and the set S of all countably infinite seq

Real Life Applications of Complex or Imaginary Numbers

When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Question:

Asymptotes, Synthetic Division and Even and Odd Power Theorem

1. FIND ALL VERTICAL ASYMPTOTES OF THE FUNCTION. *******X+5 F (x) ------------------- ****4x squared+7 2. WHICH SHOWS THE TRUE STATEMENT FOR THE GRAPH OF THE RATIONAL FUNCTION g. *******X+2 g(x) ------------------- ****x squared+2x-3 3.USE SYNTHETIC DIVISION TO FIND UPPER AND LOWER BOUNDS OF THE REAL ZEROS OF f.

Riemann Sums, Taylor Polynomials, Taylor Residuals

Please see the attached file for the fully formatted problems. 1. ? Calculate the Taylor Polynomial and the Taylor residual for the function . ? Prove that as , for all . ? Find the Taylor series of f. ? What is the radius of convergence for the Taylor series? Justify your answer. 2. ? Let f:[0,1] be a bo

Polynomials

A. Solve the following questions involving fundamental operations on polynomials a.Find p(x) + 4q(x) given p(x)=4x4 + 10x3 - 2x2 + 13 and q(x) = 2x4+ 5x2 - 3 b. Find P(-1/2) if P(x) = 2x4 + x3 + 12 c. Simplify: (-4 + x2 + 2x3) - (-6 - x + 3x3) - (-6y3 + y2) d. Add: (2x2 + 6y2 + 4z2 + 3xy + yz + zx) + (4x2 + 3y2

Euler Totient Function (Six Problems)

For this problem it helps to know that: 3x7x13 = 273 (a) Define the Euler Totient function, (SYMBOL) For (b) to (f) please see attached. (PLEASE SEE ATTACHMENT FOR COMPLETE PROBLEM AND PROPER SYMBOLS)

Polynomial Example Problems

Ace manufacturing has determined that the Cost of Labor for producing x transmissions is: 0.3x(squared) + 400x+550 dollars While the Cost of Materials is: 0.1x(squared) + 50x+800 dollars · Write a polynomial that represents the total cost of Materilas and Labor for producing x transmissions. · Evaluate the total cost pol

Proof : Polynomials

Problem: Let f(x) and g(x) be nonzero polynomials in R[x] and assume that the leading coefficient of one of them is a unit. Show that f(x)g(x) doesn't equal 0 and that deg[f(x)g(x)] = deg(f(x)) + deg(g(x))

Number Theory

Find all the integers such that when the final digit is deleted the new integer divides the original one. Can you generalize this to deleting other digits?

Approximation of Integrals and Taylor Polynomials

Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square] dx, as x from 0 to 1 ), (a) Verify that ( LIMIT (n/ln(n)) [INTEGRAL (ln square (x)dx) - (ln square (n))] as n approach to the infinity ) = 1 (b) Compute LIMIT ((n square)/

Approximation with Taylor Polynomials: Example Problems

Note:% just a symbol Let I be an open interval containing the point x.(x not), and suppose that the function f:I->R has a continuous third derivative with f'''(x)>0 for all x in I. Prove that if x.+h is in I, there is a unique number % = %(h) in the interval (0,1) such that f(x.+h) = f(x.) + f'(x.)h + f"(x.+%h)(h^2)/2

Differentiation of Polynomials : Proofs

Please see the attached file for the fully formatted problems. Let g be a function which can be differentiated four times on the interval [-1,1]. Denote . 1) Show that when g is a polynomial of degree less than or equal to 3. 2) Let P be the interpolation polynomial of f at the points -1, , , 1. a) Show that . b

Primes and divisibility word problem

When the accountants for lose-a-digit Computer, Inc. had finished preparing their annual budget, they presented the final figures to the president, I.M. Smart. "It looks like a good year," he exclaimed. "The amount of the budget just happens to be the smallest number of cents (other than one cent) that is a perfect square, a per

Modern algebra

(a) Let G = GL(2,R) be the general linear group. Let H=GL(2,Q) and K= SL(2,R) = {A is an element of G: det (A) =1} Show that H,K are subgroups of G (b) Let p be a prime number and a is an element of Z. Prove that a^p ,is equivalent to, a mod p

Working with natural numbers and divisibility.

The natural number 28A9B consists of different numbers and A is not equal to 0. When the number is divided by 9 the remainder is 7 and when it is divided by 5, the remainder is 1. What is A-B=?