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Number Theory

Classifying polynomials as monomials, binomials and trinomials

After completing please Classify the 15 given polynomials as monomials, binomials, trinomials, and polynomials. Use the format given below for categorizing the polynomials. Remember to simplify BEFORE you classify! - a. 2x3+3x2 + 2x - 1 b. ax4 - bx2 c. 3a4 - 6a3 + 3a2 d. -4xyz e. 3ab - 7cd + 5ac f.

Factoring Expressions and Solving Equations

Please see the attached file for the fully formatted problems. 1. Factor completely. 25x2 + 80x + 64 A) Prime B) (5x + 8)2 C) (5x + 8)(5x - 8) D) (5x - 8)2 2. Factor completely. 15z2 + 14z - 8 A) (15z + 4)(z - 2) B) (3z - 4)(5z + 2) C) Prime D) (3z + 4)(5z - 2) 3. Write the expression in lowest t

C++ - polynomials implementation

Design and implement a class for dealing with polynomials. The polynomial a(n)x^n + a(n-1)x^(n-1) + . . . + a0 will be implemented as a linked list. Each node will contain and int value for the power of x and an int value for the corresponding coefficient. The polynomial operations should include addition, multiplication, an

Rational function and polynomials

The reason why polynomials are so important is that there is a theorem from Analysis that says that any continuous function defined on an interval of the real line can be approximated arbitrarily closely by a polynomial. So polynomials are useful to ¿model¿ any kind of function on a closed interval. However, polynomials ¿ge

Polynomials and rational function

Now consider a rational function, which is the ratio of two polynomials. These two polynomials will each have a set of zeros, and note that at a zero of the denominator we are actually dividing by zero. The zeros of the denominator are called poles, and they are point where the rational function becomes infinite (unless there

Exponents and Polynomials

Simplify. All variables represent nonzero real numbers. 1.) -5y^4 (y^5)^2 15y^7 (y^2)^3 2.) 3y^8 This whole problem is ^4 2zy^2 3.) 3^3 This whole problem is ^4

Time Sequence diagrams

Please reference this figure for the question: 55. Draw 3 time sequence diagrams that illustrate the flow of frames between points B and G in Figure 1 using the following information: a. Stop and Wait ARQ, SENDER sends four frames (F1, F2, F3, and F4). F1, F2 and F4 are received error free, but F3 is lost in the

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

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

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^

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 =

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

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

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

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)

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

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.

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