Answers are below each question.
I’ll be using the ^ sign for exponents. But you have to use your tool. So x^2 means you enter x2.
0
-2, 2
64 w^2 y^3
3
Remember that I use ^ for exponents.
q+4
q-4
5/3
3 r^3 a^7
5
v+1
v-1
d/f
(z+8)(z+1)
z
6/5
x+2
x-2
18z^7
(z-5)(z+5)(z+3)
First box = s
Second box = s+2
Third box = 4
21x(x+4)(x+2)
10
4+w
2x+3
x-8
49+2z
28z^3
12u
u^2 - 4
t^2 + 15t
t^2 - 49
Choice C
3w-21
11w
9-8r
3r^2-3r
10fb – f^2 – b^2
f^2 – b^2
6v-6
v-8
15
w+8
y
y-3
Polynomials
Retail companies must keep close track of their operations to maintain profitability. Often, the sales data of each individual product is analyzed separately, which can be used to help set pricing and other sales strategies.
Application Practice
Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Save As from the Filemenu. You must show your work to receive credit.
Questions / Instructors Comments1. In this problem, we analyze the profit found for sales of decorative tiles. A demand equation (sometimes called a demand curve) shows how much money people would pay for a product depending on how much of that product is available on the open market. Often, the demand equation is found empirically (through experiment, or market research).
1a. Suppose a market research company finds that at a price of p = $20, they would sell x = 42 tiles each month. If they lower the price to p = $10, then more people would purchase the tile, and they can expect to sell x = 52 tiles in a month’s time. Find the equation of the line for the demand equation. Write your answer in the form p = mx + b. Hint: Write an equation using two points in the form (x,p). Provide your answer and show your work below. ( 3 points)
A company’s revenue is the amount of money that comes in from sales, before business costs are subtracted. For a single product, you can find the revenue by multiplying the quantity of the product sold, x, by the demand equation, p.
1b. Substitute the result you found from part a. into the equation R = xp to find the revenue equation. Provide your answer in simplified form and show your work below. ( 3 points)
R = x(-x + 62)
R = -x2 + 62x
The costs of doing business for a company can be found by adding fixed costs, such as rent, insurance, and wages, and variable costs, which are the costs to purchase the product you are selling. The portion of the company’s fixed costs allotted to this product is $300, and the supplier’s cost for a set of tile is $6 each. Let x represent the number of tile sets.
1c. If b represents a fixed cost, what value would represent b? Provide your answer below ( 2 points).
B represents the fixed costs, which are 300.
1d. Find the cost equation for the tile. Write your answer in the form C = mx + b. Provide your answer below ( 2 points).
Plug in the values given in the problem, using slope intercept form:
C = 6x + 300
The profit made from the sale of tiles is found by subtracting the costs from the revenue.
1e. Find the Profit Equation by substituting your equations for R and C in the equation . Simplify the equation.Provide your answer below ( 4 points).
P = R - C
P = -x2 + 62x - (6x +300)
P = -x2 + 62x - 6x - 300
P = -x2 + 56x - 300
1f. What is the profit made from selling 20 tile sets per month?Provide your answer below ( 2 points).
Plug in x = 20:
-20^2 + 56*20 - 300
P = 420
1g. What is the profit made from selling no tile sets each month?Interpret your answer in the space below. (This means providing explanations in complete sentences.) ( 2 points).
Plug in 0:
-02 + 56*0 - 300
= -300
They have to pay the fixed costs of $300, regardless of whether there are any sales. Therefore, there is a loss of 300 dollars.
1h. Use trial and error to find the quantity of tile sets per month that yields the highest profit. Provide your answer and show your work below ( 3 points).
Starting with x = 25, and trying different values:
x = 25, p = $475
x = 26, p = $480
x = 27, p = $483
x = 28, p = $484
x = 29, p = $483
x = 30, p = $480
They should sell 28, to make the most profit.
1i. How much profit would you earn from the number you found in part h? Provide your answer and show your work below ( 3 points).
The profit is $484, when they sell 28, as determined in the previous part.
1j. What price would you sell the tile sets at to realize this profit? Hint: Use the demand equation from part a. Provide your answer and show your work below ( 3 points).
Remember the demand function:
p = -x + 62
Plug in x = 28:
p = -28 + 62
p = $34
1k. The break evenvalues for a profit model are the values for which you earn $0 in profit. Use the equation you created in question one to solve P = 0, and find your break even values. Provide your answer and show your work below ( 3 points).
Set P to 0:
P = -x2 + 56x - 300
0 = -x2 + 56x - 300
Factor this equation:
0 = (x-6)(x-50)
Setting each term to 0 gives:
x = 6 and 50
- In 2002, Home Depot’s sales amounted to $58,200,000,000. In 2006, its sales were $90,800,000,000.
For 2002:
5.82*1010
For 2006:
9.08*1010
You can find the percent of growth in Home Depot’s sales from 2002 to 2006 by following these steps:
- Find the increase in sales from 2002 to 2006.
- Find what percent that increase is of the 2002 sales.
Find the increase in sales from 2002 to 2006.
(9.08 - 5.82 ) * 1010
= 3.26 * 1010
Find what percent that increase is of the 2002 sales, using division.
3.26*1010 / (5.82*1010)
= 0.56 = 56%
The Home Depot, Inc. (2007, March 29). 2006 Annual Report. Retrieved from
3. A customer wants to make a teepee in his backyard for his children. He plans to use lengths of PVC plumbing pipe for the supports on the teepee, and he wants the teepee to be 12 feet across and 8 feet tall (see figure below). How long should the pieces of PVC plumbing pipe be? Provide your answer and show your work below ( 5 points).
Using the Pythagorean theorem:
For these written ones, please don’t forget to rewrite them in your own words
your responses (in your own words) to the following three items
1.(2.5 points) Here is a number game: Take any number (except for 1). Square that number and then subtract one. Divide by one less than your original number. Now subtract your original number. Did you reach 1 for an answer? You should have. How does this number game work? Hint: Redo the number game using a variable instead of an actual number and rewrite the problem as one rational expression. How did the number game use the skill of simplifying rational expressions?
Try it with 3.
Square it: 9.
Subtract 1: 8
Divide by 1 less than the original number: 8/2 = 4
Subtract the original number: 4-3 = 1
We got to 1!
With a variable…
Start with "x".
Square it: x^2
Subtract 1: x^2 - 1
Divide by 1 less than the original number (which is x-1): (x^2-1)/(x-1)
Factor the numerator:
(x+1)(x-1)/(x-1)
Cancel like terms:
x+1
Subtract the original number "x": x+1-x
= 1
We got to 1!
2.(1.25 points) Create your own number game using the rules of algebra and post it for your classmates to solve. Think about values that may not work.
Start with any number, except -3.
Add 1.
Square the result.
Subtract 4.
Divide by 3 more than the original number.
Add 1.
You should be back to where you started!
Here is how it works:
Start with any number. “x”
Add 1: x+1
Square it: (x+1)^2 = x^2 + 2x + 1
Subtract 4: x^2 + 2x – 3
Factor: (x+3)(x-1)
Divide by 3 more than the original number: (x+3)(x-1)/(x+3) = x-1
Add 1: x-1+1 = x
We are back to where we started!
3.(1.25 points) State whether your number game from item 2 uses the skill of simplifying rational expressions.
Yes, it uses the skill of simplifying rational expressions. When we divided by 3 more than the original number, you are simplifying the rational expression.
1.(2.5 points) How is doing operations—adding, subtracting, multiplying, and dividing—with rational expressions similar to or different from doing operations with fractions? Can understanding how to work with one kind of problem help you understand how to work another type? When might you use this skill in real life?
Performing operations on rational expressions is very similar to doing those same operations on regular numerical fractions, since rational expressions are just fractions with variables in them. In order to add or subtract, you need to make sure you have a common denominator. To multiply, you multiply the numerators and denominators separately and then make a new expression. To divide, you flip the second fraction and then you multiply. The main difference between the two types is that rational expressions can have variables and exponents in them.
If you can do operations on one type, you can easily learn how to do the other type, since they are indeed the same skills of math.
You can use simplifying rational expressions in real life to figure outhow long it would take two people working together to mow a lawn, given the amount of time it takes each person alone.
2.(1.25 points) How do we find the greatest common factor of a polynomial? Demonstrate the process with an example, showing your work.
To find the greatest common factor of a polynomial, you have to find the greatest common factor of the coefficients, and then separate gcf’s for each of the variables. The gcf for a given variable is the lowest power in any of the terms (or 1 if the variable doesn’t appear in one of the terms). Once we have this, we put it back together to form a new expression.
For example:
GCF(6x^8, 9x^3)
First, get the gcf of the numbers: 3
Then of the x’s, which is the lowest power: x^3
Put it together giving:
3x^3
- (1.25 points) When finding the greatest common factor of a polynomial, can it ever be larger than the smallest coefficient? Can it ever be smaller than the smallest coefficient?
The greatest common factor can’t be larger than the smallest coefficient (assuming they are positive), since then it couldn’t possibly be a factor of that coefficient. For example, if the coefficient is 2, then 4 can’t be a factor of that. (This assumes that the coefficients are positive. If they are negative, then it can happen like this: GCF(-4, -3) = -1, and -1 is larger than either of the coefficients.)
It can definitely be smaller than the smallest coefficient. In many cases, the GCF is just 1, if there is no common factor other than 1. For example, if the coefficients are 3 and 7, then the GCF is 1.