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# Basic Algebra

### Proof by Induction: Example Problem

Prove by induction: If a > 0 then 1/a > 0, where a is an integer.

### Finding the finite formula for the infinite sum

1) When is sin(x)=0? Why is it plausible that sin(x)=x(1-x/pi)(1+x/pi)(1-x/2*pi)(1+x/2*pi) ... ? 2) Explain how Euler concludes from 1) that pi^2/6=1+1/4+1/9+1/16 ... . 3) Compare the formula in 2) to Archimedes' calculation for pi. Which of these methods is more efficient and why? 4) Why does the argument from 3) not w

### Unique Linear Polynomial that Contains Two Points

a. Consider the simplest case of a linear polynomial f(x) = mx + b. Suppose f(?) = 0 and f(0) = ?. Does this information allow you to find m and b? Explain. b. Now we look at a quadratic function f(x)=ax^2 +bx+c. Suppose ?1 does not equal ?2 and f(?1) = f(?2) = 0. How much does this information allow you to conclude a

### simplifying the square and finding values

2. simplify the square of 96b(cubed)/144a(4th power) 3. Find the value(s) of b that would make the following true: 5b(squared)-125=0. 4. Solve using the quadratic formula: 2x(squared)+2x=3. 5. State the number of zeroes for each equation: x(squared)=4x x(squared)+9=6x x(squared)+25=0 6. a mode

### The Exact Value of the Expressions

Find the exact value to the expressions cos(alpha+beta) and sin(alpha+beta) and tan(alpha+beta) under the following conditions.. cos(alpha)=21/29, alpha lies in quadrant 4, and sin(beta)= -4/11, beta lies in quadrant 3.

### Present Value: Jim Nance has been offered a future payment

Present Value: Jim Nance has been offered a future payment of \$500 three years from today. a. If his opportunity cost is 7% compounded annually, what value should he place his opportunity today? b .What is the most he should pay to purchase this payment today? c.what is the most he should pay to purchase this payment tod

### Factor the following trinomial completely.

5. Factor c^2 + 6c + 9. 6. Factor w^2 - 8w + 16. 7. Factor the following trinomial completely. x^2 - 0.2x + 0.01 8. Factor the trinomial completely. 2x^2 + 8x +8 9. Factor 16c^2 - 40ct + 25t^2. 10. Factor the trinomial completely. - 48h^3 + 72h^2y - 27hy^2

### Algebra

3. In 2002, Home Depotââ?¬â?¢s sales amounted to \$58,200,000,000. In 2006, its sales were \$90,800,000,000. a. Write Home Depotââ?¬â?¢s 2002 sales and 2006 sales in scientific notation. You can find the percent of growth in Home Depotââ?¬â?¢s sales from 2002 to 2006 by following these steps: ââ?¬¢ Find

### What is the effective annual rate?

The Baxter, Inc. borrowed \$100,000 for six months from the bank. The rate is prime plus 2 percent. The prime rate was 8.5 percent at the beginning of the loan and changed to 9 percent after two months. This was the only change. How much interest must The Jackson Corporation pay? What is the effective annual rate?

### total overtime wages

Consider the following aggregate production planning problem: January February Dema Demand 100 units 120 units Produ Production plan

### Profit equation as a function of output using demand equation

Mickey Mouse has a trap company with fixed costs of \$89,000 and operating costs of \$50 per trap and a market demand curve of: price=950-.10(output). Since Mick can't set his price find the revenue equation for his mousetrap business. a) Construct the profit equation as a function of output using the demand equation. What doe

### The Malthusian and Logistic Models of Population Growth

Please see the attached file. This is one question but broken in parts. Formula for Malthusian Model: dP/dt = r*P Formula for Logistic Model: dP/dt=rP(1-P/K) PS> I Prefer hand written work showing the steps if it is neat. It's up to you..I just have a hard time reading fractions & exponents in a word document etc.

### In what ways can the cycle of inequality be broken in schools?

Savage Inequalities is a wonderful book to reference here as it does point to the problem of inequitable funding. In other words the poor stay poor because the school is funded disproportionally from community to community. This then impacts society's views of certain minorities and the poor, but then it also continues to fuel t

### Problem Solving: Example Algebra Problem

Alice watched an ant hunting for food. The ant was traveling in a straight line toward a crumb on the floor. Suddenly, it turned around and crawled 8 inches back to Alice's shoe. Another ant joined it and together they hurried 13 inches back toward the crumb. The two ants stopped and came back 4 inches toward Alice's shoe. Final

### Calculate the value of given bonds.

Question 1 Pacific Homecare has three bond issues outstanding. All three bonds pay \$100 in annual interest plus \$1,000 at maturity. Bond S has a maturity of five years, Bond M has a 15-year maturity, and Bond L matures in 30 years. a. What is the value of these bonds when the required interest rate is 5 percent, 10 per

### solution

Question 19 of 19 4.0 Points The ABC Cement Corporation produces three varieties of high quality cements that are supplied to construction companies. The annual demand for all these varieties and their corresponding probabilities for the year 2010 are as follows. the year 2010 are as follows. Variety Demand (in tons) Probabil

### System of Equation

The Soccer Boosters club sells hotdogs and hamburgers at the games to raise funds for the soccer team. Chili Cheese Hotdogs sell for \$3.00 and hamburgers sell for \$2.50. At the game last week 175 people purchased a chili cheese hotdog or hamburger and exactly \$480 was collected. How many of each was sold?

### Algebra word problem

The butcher, the baker, and the candlestick maker purchased a lottery ticket together and won \$36,000 in the weekly lottery drawing. Since the three friends each have different risk tolerances and investment strategies, they decided that each of them would receive a certain amount of money to invest for the future. The butc

### Basic Algebra: Polynomial, Exponent and Percentage

See attachment and please show your work. 1. Add (7x^3 - 6x^2 + 8x) + (6x^3 + 8x^2 -13) 2. Subtract. (2v^5 + 7v) - (-8v^5 - 6v + 7) 3. Subtract. (8x^4 + 4x^3 - 1) - (9 x^2 - 3x + 8) 4. Find the hole using the percent equation. 35 patients is 50% of what number of patients? 5. Use the product rule to simplify the ex

### Assess auto loans.

1. For the purpose of this exercise, you can ignore sales tax. 2. Determine the annual interest rate for your loan using information from a local bank or an internet ad. Reduce this rate by 1%. This is r expressed as a decimal. 3. Decide the time, in years, you wish to repay the loan (typically, 3-7 years, half years are ok).

### Determining Payments for a Loan

Eric is paying \$39 per week to pay off a loan for \$14,568.  How many weeks will he have to make payments to pay off the loan? How much will his last payment be?

### Finding Linear Equations and Functions

1. Find the equation of a line that contains the point (3, -2) and has an undefined slope. 2. Find the equation of the line perpendicular to the line 2x + 3y =4 through the point (2, -6). Write the answer in standard form. 3. Graph the following: y ≥ x + 1 4. Let f(x) = 2x-3 and g(x) = . Evaluate the following: f(1)

### determine the empirical probability that she receives an A

Mr. Doodleâ??s grade distribution over the past 3 years for a course in college algebra is shown in the chart below. Grade Number A 45 B 180 C 210 D 95 F 65 I 5 If Jane plans to take a college algebra course with Mr. Doodle, determine the empirical probability that she

### Problems Involving Free Groups and Free Products

a. Show that in the free product, the center Z (G_1 x G_2) is trivial if |G_1| > 1 and |G_2| > 1. b. Determine the elements of finite order in G_1 x G_2. c. Show that the free group on a set X has no elements of finite order ( other than the identity).

### Integral Domains and Principal Integral Domains

See attached file. I am having problems with a couple of abstract algebra problems regarding integral domains and principal integral domains (PIDs). A commutative ring satisfied the DCCP if <a1> ⊇ <a2> ⊇ <a3> ⊇ ... implies that an ~ an+1 ~ ... for some n ≥ 1. Show that an integral domain R has DCCP if and only

### Branch cuts points of log

Can you please wirte down a precise method to find the branch cuts and points of log(f(z)) for any f(z)? like what is the difference between log(z^2-1), log(1-z^2) and log(1-1/z^2) and how can you tell?

### branch points

When you're given Log(f(z)) of any kind of f(z), is there a particular method to find the domain of analyticity of Log(f(z)), how can you find its branch cuts? I'm stuck with the following Log(1-1/z^2), i don't understand how to prove that the branch cut is [-1,1]

### Sketching Graph Function Calculators

1. If f(x) = x^2 - 3x determine the following (i) f(-2) (ii) f(-2a) (iii) f(2-a) 2. (i) Use your calculator to sketch a graph of the function f(x) = 2/sqrt(9 - x^2) (ii) Explain the restrictions to the domain of the function. (iii) Name and describe the feature of the graph th

### Number and Inequalities

Question 2 Number and Inequalities 1.A cylindrical vessel with one end open is made from a given piece of material. Show that its volume is greatest if the height and radius are equal. 2. a,b,c are non zero rational numbers. Show that if: ³√a + ³√b + ³√c = 0 And if none of ³√a , ³√b , ³√c is rat