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# Number Theory

### 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

### Functions : Inverse Function Composition, Polynomials and Exponential Growth

Please see the attached files for the fully formatted problems.

### 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

### Describe an application where long division can be used in day to day life.

Describe an application where long division can be used in day to day life.

### Polynomials and Complex Roots

State how many complex and real zeros the functon has. x^2 -2x+7 x^4-2x^2+3x-4 find all of the zeros and write a linear factorization of the function f(x)=x^3+4x-5 r(x)=3x^4+8x^3+6x^2+3x-2 using the given zero, find all of the zeros and write a linear factorization of f(x) 1+i is a zero of f(x)=x^4-2x^3-x^2+

### Finding Polynomials From Complex Roots

Write a polynomial function of minimum degree in standard form with real coeffiecients whose zeros include those listed. 6. 1-2i and 1+2i 12. -2 and 1+2i Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed. 15. 2(multiplicity2), 3

### Prove that if p is a prime number p &#8800; 3, then 3 divides p^2 + 2.

College level proof before real analysis. Please give formal proof. Please explain each step of your solution. Thank you.

### 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.

### Synthetic Division and Finding Polynomials

Please see the attached file - only: #25, 27, 29, 31

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

### Synthetic Division and Polynomials

1. Explain what synthetic division is. Illustrate with example. In synthetic division, what relationship does the divisor and remainder have to the original polynomial function, according to the Remainder Theorem? 2. What general characteristics for the graph of any polynomial function can be found by looking at its equation

### polynomials, identify the polynomials of degree one.

From the given polynomials, identify the polynomials of degree one x3 + 4x + 8 76 + 5x (x)1/2 + 5x - 6 500 + (45)1/2x 289y + 6 -76y (37)1x1 (9x2)1/2 + 3 89x4 + 3y + 5

### 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

### Subtract the Polynomials

Need assistance to understand the following. Please see attached file.

### Modular Arithmetic Functions

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

### 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)

1) (-2)(-2)x+Â¹ 2) -2(3xÂ²yÂ³)Â² 3) -2Â²(3xyÂ²)Â³

### 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 equal to 2. Prove that L_1 + 2L_2 + 4L_3 +8L_4 + ... + 2^(n - 1) L_n = 2^n F_(n + 1) - 1

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_(

### Numerical Theory : Find the Pattern

Find the Pattern: 7, 28, 24, 45

### Finding a Quotient

Please see the attached file for the fully formatted problem.

### Maclaurin Polynomials

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 P&#959; (&#959; 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,

### Problems with Taylor polynomials

First, find the Taylor polynomial Pn (n is a subscript) of order n for the function f with base point X&#959; (&#959; 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, X&#959; (&#959; is a subscript) = 0

### Express f(x) as a product of linear and quadratic polynomials with real coefficients.

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.

### 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?

### Polynomial Long Division : Linear Divisor

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.

### 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