Post a 50-word response to the following: How do you determine if a polynomial is the difference of two squares?
A polynomial is a difference of two squares if it has exactly two terms. The terms need to have a negative sign between them. Each of the terms must themselves be a perfect square like 4 or 16x^2 or 256x^8. If it meets all of those requirements, then it's a difference of squares, an example being 16x^2 - 256x^8.
Appendix:
1. In this problem, we will 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).
a. Suppose that 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)).
Slope = rise/run
= (42-52)/(20-10)
= -10/10
= -1
p-p1 = m(x-x1)
p-52 = -1(x-10)
p - 52 = -x + 10
p = -x + 62
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.
b. Substitute the result you found from part a into the equation R = xp to find the revenue equation. Provide your answer in simplified form.
R = x(-x + 62)
R = -x^2 + 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.
c. If b represents a fixed cost, what value would represent b?
The fixed costs are 300.
d. Find the cost equation for the tile. Write your answer in the form C = mx + b.
Plug in the values given in the problem:
C = 6x + 300
The profit made from the sale of tiles is found by subtracting the costs from the revenue.
e. Find the Profit Equation by substituting your equations for R and C in the equation . Simplify the equation.
P = R - C
P = -x^2 + 62x - (6x +300)
P = -x^2 + 62x - 6x - 300
P = -x^2 + 56x - 300
f. What is the profit made from selling 20 tile sets per month?
Plug in x = 20:
-20^2 + 56*20 - 300
P = 420
g. What is the profit made from selling 25 tile sets each month?
Plug in 25:
-25^2 + 56*25 - 300
P = 475
h. What is the profit made from selling no tile sets each month? Interpret your answer.
Plug in 0:
-0^2 + 56*0 - 300
= -300
Those are the fixed costs that have to be paid regardless of sales.
i. Use trial and error to find the quantity of tile sets per month that yields the highest profit.
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.
j. How much profit would you earn from the number you found in part i?
The profit is $484, when they sell 28.
k. What price would you sell the tile sets at to realize this profit (hint, use the demand equation from part a)?
Remember the demand function:
p = -x + 62
p = -28 + 62
p = $34
2. The break even values 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.
Set P to 0:
P = -x^2 + 56x - 300
0 = -x^2 + 56x - 300
0 = (x-6)(x-50)
x = 6 and 50
3. In 2002, Home Depot's sales amounted to $58,200,000,000. In 2006, its sales were $90,800,000,000.
a. Write Home Depot's 2002 sales and 2006 sales in scientific notation.
5.82*10^10
9.08*10^10
b. What was the percent growth in Home Depot's sales from 2002 to 2006? Do all your work by using scientific notation.
You can find the percent of growth in Home Depot's sales from 2002 to 2006, follow these steps:
- Find the increase in sales from 2002 to 2006.
(9.08 - 5.82 ) * 10^10 - = 3.26 * 10^10
- Find what percent that increase is of the 2002 sales.
3.26*10^10 / (5.82*10^10)
= 0.56 = 56%
(source: Home Depot Annual Report for FY 2006:
4. 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). How long should the pieces of PVC plumbing pipe be?
a^2 + b^2 = c^2
(12/2)^2 + 8^2 = c^2
6^2 + 8^2 = c^2
c = sqrt(6^2 + 8^2)
c = sqrt(100)
c = 10 feet
Please note that I use ^ for exponents. But you have to use your tool. So x^3 means x3.
Number 1:
Not posted
Number 2:
6ab
Number 3:
5x(x+7)
Number 4:
5xy (x^7y^5 + 6x^5y^4 + 5)
Number 5:
(x^6 + 2)(x+5)
Number 6:
(3x^2+4)(5x-7)
Number 7:
(x^2-2)(x+4)
Number 8:
(c+5)(c+4)
Number 9:
(t-8)(t-7)
Number 10:
(t-8)(t+6)
Number 11:
(b-6f)(b+4f)
Number 12:
(b-2)(5b+1)
Number 13:
(5c-7)(4c+9)
Number 14:
-3(3c+2)(4c-5)
Number 15:
(2s-3)(3s+1)
Number 16:
(r+4)(r+6)
Number 17:
(6v-7)(v+6)
Number 18:
(4s-5)(5s+2)
Number 19:
2(3d-2)(d+4)
Number 20:
(3g+2d)(2g+3d)
Number 21:
Yes
Number 22:
(r+6)(r+6)
Number 23:
3(r-4)(r-4)
Number 24:
(r-7)(r+7)
Number 25:
w(4-7w)(4+7w)
Number 26:
(0.6y+0.01)(0.6y-0.01)
Number 27:
Choice B
Number 28:
(a+3)(a^2 – 3a + 9)
Number 29:
2(2r^2-d^2)(4r^4 + 2r^2 d^2 + d^4)
Number 30:
5(v^2+25)(v+5)(v-5)
Number 31:
-4, -49
Number 32:
0, -4
Number 33:
3, -5/9
Number 34:
-6, -5
Number 35:
0, 5
Number 36:
-5, 3
Number 37:
Width = 3
Length = 12
Number 38:
110
Number 39:
11, 12