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Number Theory

Polynomials Factored Completely

Factor completely (see attached) Factor Completely a. b. c. d. (11x-6y) (11x + 6y ) Simplify a. b. c. d. Multiply a. b. c. d. Solve this system of equations. 7x - 5y = 1

Written Numbers in Expanded Forms

Please assist with attached math problems. 1 Write this number in expanded form a. 247,089 b. What digit tells the number of thousands c. What digit tell the number of ten thousands 2 Write this number in words $8,886 3 Name the property of addition and explain your choice. a. 3+(0 + 6) = (3 + 0) + 6 b. 0 + a

Problems with a Set Theory in Real Analysis

Undergraduate senior level Real Analysis. Please show me formal math proofs. Show that Q(set of rational numbers) ~ N(set of natural numbers). (Suggestion from my professor: prove by letting set Qn < Q and a/b (a and b are some integers, but b is not zero) and Q = union of Qn (Qn = Q1, Q2, Q3, ....) .)

A Set Theory in Real Analysis

Formal Math Proofs Prove that each of the following sets is countable: a) The set of all numbers with two distinct decimal expansions (like 0.500... and 0.4999...); b) The set of all rational points in the plane (i.e., points with rational coordinates); c) The set of all rational intervals (i.e., intervals with ratio

Marriage penalty eliminated ...

Marriage penalty eliminated. The value of the expression 4220 + 0.25(x - 30,650)is the 2006 federal income tax for a single taxpayer with taxable income of x dollars, where x is over $30,650 but not over $74,200. a) Simplify the expression. b) Find the amount of tax for a single taxpayer with taxable income of $40,000. c) W

Regular bipartite graph

Consider the sets A0 := {0, 1, 4}, B0 := {0, 2, 8}. Consider the sets Ai := A0 + i := {i, i + 1, i + 4} ,and Bi := B0 + i := {i, i + 2, i + 8}, for i = 1, 2, . . . , 12. All addition here is performed modulo 13. Consider the bipartite graph G whose vertices are the sets A := {Ai : 0 <= i <= 12} and B := {Bi : 0 <= i <= 12} (so

Exercise on Polynomials

Solve the following questions involving fundamental operations on polynomials a. Find p(x) + 4q(x) p(x)=4x^4 + 55x^3 - 23x^2 + 13 q(x)=43x^4+ 14x^2 -12 b. Find P(-1/2) if P(x) = 2x^4 + x^3 + 12 c. Simplify: (-4 + x^2 + 2x^3) - (-6 - x + 3x^3) - (-6y^3 + y^2) d. Add: (2x^2 + 6y^2 + 4z^2 + 3xy + yz + zx) + (4x^2 + 3y^

Telephoto

Using formula 1/f = 1/o + 1/i Find the image distance i for an object that is 2,000,000 mm from a 250-mm telephoto lens

Contemporary Abstract Algebra, Author: Joseph A. Gallian

Chapter 0 (Preliminaries) Q.) How would you prove the converse? A partition of a set S defines on equivalence relation on S. Hint: Define a relation as X - Y if X and Y are elements of the same subset of the partition. 10.) Let n be fixed positive integer greater than 1. If a mod n = a' and b mod n = b' .Prove that (a

Exponents, Multiplication, Division of Polynomials

See attached. 1. Evaluate the expression. Assume 2. Evaluate when y = -2 3. Evaluate when 4. Express the following using a positive exponent. Then simplify the expression . Write using a positive exponent do not evaluate 5. Express using a positive exponent = 6. Multiply and simplify =

An overview on tessellations.

An overview of the theory and results on tessellations of three types of Riemann surfaces: the Euclidean plane, the sphere and the hyperbolic plane. Roughly speaking, a tessellation of a space is a pattern which, repeated infinitely many times, fills the space without overlaps or gaps . From a mathematical point of view, tesse

Members of the Given Real Number Subset

Directions: List all numbers from the given set B that are members of the given Real Number subset. Please explain. B=[ 19, square root 8, -5, 0, 0.7 as a repeating decimal, square root of 9] Integers B= [ 17, square root of 5, -2, 0,0.7 as a repeating decimal, square root 16,] Whole numbers B= [ 6,square root v8, -1

Practice problems on Polynomials and Rational Functions

1. Explain how synthetic division may be used to find the factors/zeros of a polynomial function. Give an example of how this is accomplished. Use synthetic division to find the function value. 1) f(x) = 2x4 + 4x3 + 2x2 + 3x + 8; find f(-2). Write the quadratic function in the form y = a(x - h)2 + k. 2) y = x2 - 2x - 9

Modeling using Second and Third Order Polynomials

To explore the versatility of rational functions, choose a second-order/third-order (e.g., x2/x3) and a third-order/second-order (e.g., x3/x2) rational function. Provide a graph for the second-order rational function (e.g., x2), choosing x values in the range from -10 through +10. Then, provide at least three variations of th

Modeling using Second and Third Order Polynomials

One of the advantages of rational functions is that even rational functions with low-order polynomials can provide excellent fits to complex experimental data. Linear-to-linear rational functions have been used to describe earthquake plates. As another example, a linear-quadratic fit has been used to describe lung function after

Polynomials with Real and Complex Solutions

1. 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. 2. Find the inverse of the function f(x) = x1/3 + 2. 3. If a piece of real estate purchased for $50,000 in 1998 appr

Primes and Odd Numbers

Prove that if every even natural number greater than 2 is the sum of two primes, then every odd natural number greater than 5 is the sum of three primes. Please see the attached file for the fully formatted problems.

Simplifying Radicals

It is important to simplify radical expressions before adding or subtracting to get terms with like radicals. Once we find terms with like radicals we can then add or subtract the expressions. Simplifying the expression and obtaining terms with like radicals makes the problem less complex when solving. We can follow the same rul

Prime Numbers and Perfect Square

Solve the following problems in essay form (1 to 2 paragraphs for each problem is fine). Please show all work and show how you came up with the answer. Please make sure that you address the following questions: (1) How did you get your answer? (2) What steps did you take? (3) Where did you begin? (4) Why did you do what you di

Fitting Polynomials to Experimental Data

One of the advantages of rational functions is that even rational functions with low-order polynomials can provide excellent fits to complex experimental data. Linear-to-linear rational functions have been used to describe earthquake plates. As another example, a linear-quadratic fit has been used to describe lung function afte

Fundamental Operations with Polynomials

Solve the following questions involving fundamental operations on polynimials. Thank you 1)Find p(x)+4q(x) p(x)=3x^5+70x^3-67x^2+3 q(x)=3x^3+56x^2-19 2)Find P(-1/2)if P(x)=x^4+3x^2+2 3)Divide (6x^3-5x^2-13x+13)divided by (2x+3)= 4) Factor completely: 6x^2-28x+16 (2x-8)(3x-2) (2x-8) I need to fact

Fundamental Operations on Polynomials

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

Conducting Operations with Polynomials

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 remainder (x^4-2x^2+3)

Undecidability Theorem

According to the undecidability theorem, most software quality properties are not provable. Therefore, what kind of testing techniques do we use to achieve software quality?

Arithmetic

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

Operations for Polynomials and Vector Space

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