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Basic Algebra

Laws of Exponents and Opposites of Poynomials

1. Using one of the laws of exponents, prove that any number raised to the power 0 is 1. 2. You are given the following polynomial: 2x7 - 4x3 + 3x. If x were replaced with its opposite in each of the terms of the given polynomial, will it result in the opposite of the polynomial? Explain why or why not and illustrate to supp

Finite and Infinite Summations

a) Write an expression in sigma notation as shown for suitable terms n1, n2, and ai, for each of these finite and infinite summations. b) Use the method of telescopic sums to evaluate these finite sums, using the indicated anti-difference sequences. c) Which of the following sequences are null? Explain your answer. See

Sum of 2 nonnegative numbers is 20 Find the numbers

The sum of 2 nonnegative numbers is 20. Find the numbers if: a)if the product of one number and the square root of the other is to be as large as possible, and b)if one number plus the square root of the other is to be as large as possible.

Binary Tree Induction

# Recall that a binary tree can be defined recursively as: * A Binary Tree is either empty * or A Binary Tree consists of a node with a left and right child both of which are Binary Trees. The degree of a node in a tree is equal to 0 if both children are empty, 1 if one of the children are empty, and 2 of both children ar

Grammar Induction

Consider the grammar 1) <Number> -> <Digit><Number>|epsilon 2) <Digit> -> 0|1|2|3|4|5|6|7|8|9 Use induction to show that the number of strings in L(<Number>) of length n is equal to 10^n

Aplication of Algebraic Expressions

3. Lisa is coloring her creation. The blue color used is 50% of the green color used, yellow is 5% of the blue color used, and brown is 7% of the amount of green used. She wants to find the total amount of color used by her. a. Find the total amount of color used by Lisa using an algebraic expression. b. Comment on the kind of

Application of Algebraic Expressions

2. Mr. Pinto is a Web designer. He is trying to put 4 characters on a page. The space occupied by character 2 takes 5 units more than character 1, character 3 takes 7 units less than the character 2, and character 4 takes the same space as that taken by character 2. a. Find the total space occupied by all the characters in an a

Creating algebraic expressions

1. Three prizes are to be distributed in a Creative Design Talent Search Contest. The value of the second prize is five-sixths the value of the first prize, and the value of the third prize is fourth-fifths that of the second prize. a. Express the total value of the three prizes as an algebraic expression. b. Comment on the ki

Taylor Polynomial Real Analysis

See attachment 1) Calculate the Taylor polynomial of degree n at 0 of the function . 2) • suppose that f''(x)=0 for all . Show that f has the form f(x)=ax+b for some real numbers a,b. • let h:(-1,1) be defined by . Prove that there exists a positive integer m such that

Compound Inequality, Selling Price Range, and Retirement Pay

See the attached file. Solve each compound inequality and write the solution set using interval notation. 80) 0< 5-2x <=10 -6< 4-x<0 81) -3 < (3x -1)/ 5 < ½ 1/3 < (3-2x)/6 < 9/2 86) Selling Price Range: Renee wants to sell her car through a broker who charges a commission of 10% of the selling price. The book val

Clearly identify the variables, coefficients, and constants

Translate the following into algebraic expressions. Clearly identify the variables, coefficients, and constants: a) One fourth of one half of the product of two numbers b) Two-thirds of the quotient of two numbers c) The product of 7 and twice n


Lydia takes ten minutes more to complete an illustration than Tom. The total time taken by both of them is six hours. Form an algebraic equation to express this and identify the variables, coefficients, and constants of the algebraic expression. You are not required to solve the equation.

Measurement of Angles

1. In a certain triangle the measure of one angle is double the measure of a second angle but is 10 degrees less than the measure of the third angle. [The sum of the measures of three interior angles of a triangle is always 180 degrees.] Form an algebraic equation to express the problem and identify the variables, coefficients,

Finite Set of Embeddings

Let E be a finite extension of a field F. Show that any finite set of distinct embeddings of E into the algebraic closure of F is linearly independent over F.

Let G, H be graphs such that G is a subgraph of H. Prove or disprove each of the following: (a) alpha(G) <= alpha(H) (b) alpha(G) >= alpha(H) (c) omega(G) <= omega(H) (d) omega(G) >= omega(H)

The stability number, alpha(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that no two of the vertices in S are connected by an edge of G. The clique number, omega(G), of a graph G is the cardinality of the largest subset S of V(G), the vertex set of G, such that every pair of

Exponents and Logarithms Problems

Graph y = 2^-x for -2 <= x <= 2. Solve for x: 3^4x-5 = 27. Suppose $1000 is invested at 6 percent annual interest, compounded monthly. How many dollars wil be in the account after: a) one year? b) five years? Write in exponential form (log_9)(27) = 3/2. Write in logarithmic form: 1296 = 6^4. At 5% interest comp

ODEs with Constant Coefficients

Find the general solution of the ODEs attached (about 10 different problems involving differential equations with constant coefficients) Solve the IVPs attached.

ODE's with Constant Coefficients

See the attached file. Y1: Find the general solution of the OEDs: y'''-3y"+3y'-y=0 y"=0 y"+y'-2y=0 5y"-10'=0 And about 8 other ODEs with IVP.

Syntactic categories

Write productions that will define the syntactic category <SimpleStat>, to replace the abstract terminal simpleStat in the figure below {see attachment}. You may assume the syntactic category <Expression> stands for C arithmetic expressions. Recall that a "simple statement" can be an assignment, function call, or jump, and that

Proof about congruence modulo 43 (also expressible as equivalence modulo 43)

Let S = Z_43 (where the underscore, "_", indicates that what follows it, in this case 43, is a subscript). Let Q be a subset of S that contains ten non-zero numbers (i.e., that Q contains ten non-zero elements of S). Prove that Q contains four distinct numbers "a," "b," "c," "d" such that ab = cd in Z_43.

Lowest Common Multiple Application Word Problem

Five children collect N pieces of Halloween candy and decide to split it evenly among them. When they try to divide it they have two pieces of candy left over. One of the children leaves, taking the 26 pieces of candy she collected with her. The remaining four children try to split the N-26 remaining pieces of candy and discover