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# Basic Algebra

### How old is Karley?

Karley is twice as old as Lana. Three years from now, the sum of their ages will be 42. How old is Karley?

### Well-Ordering Axiom - Strong Induction

Prove the well-ordering Axiom by strong induction.

### Time, Rate, and Distance

I know the formula Rate x Time = Distance. However, I have a question where I am asked to find the time and not the distance? Can you give me some basic information about how to calculate the time using this formula or is there a different formula that I should be using? Thanks.

### Distance, Rate and Time Problem

Karen can row a boat 10 kilometers per hour in still water. In a river where the current is 5 kilometers per hour, it takes her 4 hours longer to row a given distance upstream than to travel the same distance downstream. Find how long it takes her to row upstream, how long to row downstream, and how many kilometers she rows?

### Solving quadratic equations by factorization.

Solve the following quadratic equations by factorization: a) x^2 - 5x = 0 b) 6t^2 = t(t-4) c) a^2 + 9a = 0

### It is a description of how to apply Mathematical Induction in proving theorem or statement. Application of mathematical Induction : F 2n+1 - Fn Fn+2 = (-1) n

Application of Mathematical Induction Application of Mathematical Induction Fibonacci Numbers :- The Fibonacci numbers are numbers that has the following properties. If Fn represents the nth Fibonacci number, F1 = 1, F2 =1, F3 =2, F4=3, F5 = 5 etc. We can find the Fibonacci number

### Exponential and Logarithmic Functions : Level of Sound

Determine the level of sound (in decibels) for the given sound intensity. (a) I=10^-3.5 watt per m^2 (jet 4 miles from takeoff) (b) I=10^-3 watt per m^2 (diesel truck at 25 feet) (c) I=10^-1.5 watt per m^2 (auto horn at 3 feet)

### Exponential and Logarithmic Functions: Example Problems

Sales and Advertising: The sales S (in thousands of units) of a product after x hundred dollars is spent on advertising is: S = 10(1-e^kx). When \$500 is spent on advertising, 2500 units are sold. (a) Complete the model by solving for K. (b) Estimate the number of units that will be sold if advertising expenditures are raised

### Irreducible Polynomial over a Field

Please see the attached file for the fully formatted problems. 5. Find an irreducible polynomial f(x) over the field Z3 with Z3[x]/(f(x)) = F243. Note that 243 = 3^5 . Please explain your reasoning and solution in as much detail as possible. Thank You.

### Proof of Inequality by Mathematical Induction

Prove that (n + 1)!>2^(n+3) for n>=3 Hint: try using mathematical induction

### Galois Groups of Irreducible Polynomials

Describe all subgroups of S5 which are Galois groups of irreducible polynomials of degree 5.

### Find Irreducible Polynomial

Find an irreducible polynomial defining the extension Q(3^1/2, 5^1/2).

### Irreducible Polynomial : Splitting Field

Let K be obtained as a field Q(alpha) where alpha is a root of P(x) = x3 &#8722;3. Find an irreducible polynomial which defines the splitting field of P(x).

### Finite Fields : Field Extensions

Please see the attached file for the fully formatted problem. Show the existence of an extension of Fq of order l for any prime l.

### Simplify the Expression

Please see the attached file for the fully formatted problem. (y2 + y - 12)/(y3 +9y2 + 20y)/[(y2-9)/(y2+3y2)]

### Determining Correlation and Distribution

6) Suppose we have a building with a floor shaped like an isosceles right triangle. The two sides adjacent to the right triangle have length 100 feet. Think of the right angle being at the origin, and other two corners at (100, 0) and (0, 100). The overhead crane is located at the origin and needs to travel to a point (X, Y), w

### Serial Process Continued

At Very Long Hotel in Florida, there are n rooms located along a very long corridor and numbered consecutively from 1 to n. One night after a party, n people, who have been likewise numbered from 1 to n, arrived at this hotel and proceeded as follows: Guest 1 opened all the doors. Then Guest 2 closed every second door beginnin

### Functions: Proof by Induction

Let n be a natural number, and let f(x) = x^n for all x are members of R. 1) If n is even, then f is strictly increasing hence one-to-one, on [0,infinity) and f([0,infinity)) = [0,infinity). 2) If n is odd, then f is strictly increasing, hence one-to-one, on R and f(R) = R. "This needs to be a proof by induction, provin

### Functions : Proof by Induction

Let n be a natural number, and let f(x) = x^n for all x are members of R. 1) if n is even, then f is strictly increasing, hence one-to-one, on [0,infinity) and f([0,infinity)) = [0,infinity). 2) if n is odd, then f is strictly increasing, hence one-to-one, on R and f(R) = R. Prove that f is strictly increasing by indu

### Central Extension Problem

Describe all nonisomorphic central extensions of Z_n by Z_2 x Z_2 meaning central group extensions of the following form 1 --> Z_n --> G --> Z_2 x Z_2 --> 1 Meaning, determine those nonisomorphic groups G that can be described by such an extension. Please also explain how you came up with the answer.

### Working with the sum of cubes.

Please see the attached file for the fully formatted problems. Problem 3: This problem itself is directly creating perceptual curiosity and at the cognitive level it is against what is known in the mathematics. The problem is: 13+23+33+.... +n3 = (1+2+3+...+n) 2 Visual representation of problem: Sum of the cubes

### Getting three thousand bananas across a one thousand mile desert

You have three thousand bananas that you have to get to a destination 1000 miles away. You can only carry 1000 bananas at a time. You also must eat 1 banana per mile for energy. Assuming you design your trip as efficiently as possible, how many bananas will you have left when you arrive at the destination? (apparently someon

### How to get a million by spelling a word whose letters' product=1000000

If we assign number values to letters in the following way: A = 26, B = 25, C = 24 and so on until Y = 2 and Z =1, spell a word such that the product of its letters is as close to a million as possible. Explain how you went about solving this problem.

### Algebraic Equations : Reciprocal Equations

As provided by E. Galua theory the general algebraic equations for a polynomial of fourth order ax^4 + bx^3 + cx^2 + dx + f=0 (*) is the maximum order type of algebraic equations the solution to which one can write down in radical expressions. Among all the equations of fourth

### Algebra : Puzzle Problems

Please see the attached file for the fully formatted problems. 1) Have you ever seen the written form of the Sanskrit language? If so, you probably are amazed at how different this ancient language from India looks from ours. Some English words, however, are based on Sanskrit. For example, cup comes from the Sanskrit work kup

### Algebra : Remainders

What is the remainder when the product of one hundred 5's is divided by 7? Please be detailed in your response.

### Algebra : Word Problem - Remainders

A large purse is full of coins. If you count them by 13's, 23's, or 31's, there will be one left over. If you count them by 73's there will be none left over. How many coins are there in the purse? Please be detailed in your response.

### Word Problem - How many Cookies?

Dad made cookies. Dad ate 1 cookie. Dave ate 1/2 dozen cookies. Kate ate 1/2 of what Dave ate. Henry & Julie each ate 1/3 of what was left. Then Jake ate 1/2 of what was left. Mom ate 1 cookie. There was 1 cookie left. How many cookies were baked?

### Algebra : Word Problem - Sums and Products

During the census, a man told the census-taker that he had three children. When asked their ages he replied, " The product of their ages is 72. The sum of their ages is my house number." The census taker turned around and ran outside to look at the house number displayed over the door. He then re-entered the house and said, "

### Algebra: Word Problem - System of Equations

Three people play a game in which one person loses and two people win each game. The one who loses must double the amount of money that each of the other players has at that time. The three players agree to play three games. At the end of the three games, each player has lost one game and each has \$8. What was the original st