Prove that if p is a polynomial with degree n and k is a real number so that p(k) = 0,

then p(x)=(x-k)q(x) where q is a polynomial of degree n-1. Must use the fact that

a^n-b^n=(a-b)(a^(n-1) + a^(n-2)*b + ... + ab^(n-2) + b^(n-1)).

karla has opened an IRA in which she deposits $250 each month that earns 3.5% interest compounded monthly

how much money will she have in the account after 25 years?

how much total money will she put into the account?

how much total interest will she earn?

Melanie is the manager of the Clean Machine car wash and has gathered the following information. Customers arrive at a rate of eight per hour according to a Poisson distribution. The car washer can service an average of ten cars per hour with service times described by an ex- ponential distribution. Melanie is concerned about the number of customers waiting in line. She has asked you to calculate the following system characteristics:

(a) Average system utilization 8/10=0.8= 80%

(b) Average number of customers in the system 8/(10-8)=8/2= 4

(c) Average number of customers waiting in line 8^2/10(10-8)= 64/20= 3.2 customers

2. Melanie realizes that how long the customer must wait is also very important. She is also concerned about customers balking when the waiting line is too long. Using the arrival and service rates in Problem 1, she wants you to calculate the following system characteristics:

(a) The average time a customer spends in the system 1/(10-8)= ½ hours

(b) The average time a customer spends waiting in line0.8/(10-8)= 0.8/2 = 0.4 hours

(c) The probability of having more than three customers in the system 0.8^3 =0.512

(d) The probability of having more than four customers in the system 0.8^4 =0.4096

If Melanie adds an additional server at Clean Machine car wash, the service rate changes to an average of 16 cars per hour. The customer arrival rate is 10 cars per hour. Melanie has asked you to calculate the following system characteristics:
(a) Average system utilization
(b) Average number of customers in the system

(c) Average number of customers waiting in line

Melanie is curious to see the difference in waiting times for customers caused by the additional server added in Problem 3. Calculate the following system characteristics for her :

(a) The average time a customer spends in the system

(b) The average time a customer spends waiting in line

(c) The probability of having more than three customers in the system

(d) The probability of having more than four customers in the system

After Melanie added another car washer at Clean Machine (service rate is an average of 16 customers per hour), business improved. Melanie now estimates that the arrival rate is 12 customers per hour. Given this new information, she wants you to calculate the fol- lowing system characteristics:

(a) Average system utilization

(b) Average number of customers in the system

(c) Average number of customers waiting in line

As usual, Melanie then requested you to calculate sys- tem characteristics concerning customer time spent in the system.

(a) Calculate the average time a customer spends in the system.

(b) Calculate the average time a customer spends waiting in line.

(c) Calculate the probability of having more than four customers in the system.

Calculate the principal value of (i(i − 1))^i AND the principal value of (i^i)(i − 1)^i.

On 2002/4/1, Peter borrowed $2000, agreeing to pay interest at 6%/year compounded monthly. He paid $400 on 2004/9/1 and $500 on 2008/11/1. He will make two more payments on 2011/10/01 and 2013/7/01, with the payment on 2011/10/01 being 20% higher than that on 2013/7/01. What payment will he make on 2013/7/01? Remark: Dates are given in the format YYYY/MM/DD.

Show that if y is a reparametrization of a curve x, then x is a reparametrization of y.

Give an example of a math problem that you have seen in a previous math class that would not be considered a discrete math problem. In order to receive full credit for this question you must: state the problem, and give a brief explanation as to why the problem is not a discrete math problem. You DO NOT have to give the solution to the problem

Prove that the rank of the Jacobian matrix does not depend on the choice of the generators of the ideal.

South Shore Construction builds permanent docks and seawalls along the southern shore of Long Island. Although the firm has been in business only five years, revenue has increased from $308,000 in the first year of operation to $1,084,000 in the most recent year. The following data show the quarterly sales revenue in thousands of dollars: (Answer 1-3

1.Below is a simple linear regression analysis for this forecasting problem. Is this a valid model to use to forecast quarterly revenue for South Shore? Why or why not? Explain completely.

2.Based on the model in question 1, what is the forecast for Quarter 1 of Year 6?

3.Is there a seasonal trend in this data? Find the appropriate seasonal factor for Quarter 1 of Year 6 and apply it to get a new forecast for revenue in that quarter. Does this provide a better model than the one used in questions 1 and 2? Why or why not? Provide support for your answer (not necessarily numerical data).

For each of the following, subsets of R decide whether or not the set has an infimum and/or a supremum. If they exist, write down the infimum and supremum, and say whether it is an element of the set. (a) [3,7)

(b) {x ∈ Q : 1 ≤ x < 5}

(c) {1, 2, 3, . . . }

(d) {. . . , 1/4, 1/2, 1, 2, 4, . . . }

Find the mass and the center of mass of the solid E with the given density function ρ(x,y,z).

E lies under the plane z = 3 + x + y and above the region in the xy-plane bounded by the curves
y=√x, y=0, and x=1; ρ(x,y,z) = 10.

m =

x =

y =

z =

Use false position method to find the root of ?(?) = −sin(? − 5) + ? with initial guesses of 0.2 and 1. Show up to three iterations and calculate the relative percent error ?? for each iteration possible? Show full details for at least one iteration to get full points. Also, if three significant figure accuracy is required, show if the value after third iteration is acceptable or not.

0. Lon Lee has a collection of twenty photos of friends and family that he keeps to cheer himself

   up. Each of the following questions refer to this collection.

   1. Lon wants to hang 8 of his 20 photos in a line on the wall above his desk. How many ways can he do this?

   2. How many ways can Lon mount 4 of his 20 photos in a picture frame (2 rows with 2 pictures each).

   3. Lon wants to select 9 photos of his 20 to take on a trip. How many ways can he do this?

   4. Lon has decided to give his photo collection to his three children. In how many ways can he

      partition his photo collection among this three children?

   5. Lon also has 500 dollars he wants to give his three children. In how many ways can he devide it among his three children. Let’s assume that it is OK that a child does not get any money.