1) Find the polynomial function in factored form: \(y=a(x-b)(x-c)(x-d)\) for the given data.

x | 1 | 3 | -4 | -1 |

y | 0 | 0 | 0 | 2 |

2) The Sunspot Small Appliance Company determines that the supply function for their EverCurl hair dryer is \(S(p)=6+0.001p^3\) and the demand function is \(D(p)=80-0.02p^2\), where p is the price of the hair dryer. Determine the price for which the supply is equal to the demand and the nuumber of hair dryers corresponding to this equillibrium price.

AdamTaurus
Nov 1, 2017

#1**+1 **

1) When y = 0 , x = 1, 3, or -4

A polynomial in that form with zeros at 1, 3, and -4 is

y = a(x - 1)(x - 3)(x + 4)

To find a , plug in -1 for x and 2 for y .

2 = a(-1 - 1)(-1 - 3)(-1 + 4)

2 = a(-2)(-4)(3)

2 = 24a

a = 1/12

So the polynomial is y = (1/12)(x - 1)(x - 3)(x + 4) And here's a graph to check it.

2) We want to know what p is when supply = demand . That is, when

6 + 0.001p^{3} = 80 - 0.02p^{2}

Let's solve this using a graph. The p value that makes the equations have the same y value is

p ≈ 36.27

Look at the graph or plug this value for p into either function to find out how many hair dryers are supplied and demanded at this price.

D( 36.27 ) = 80 - 0.02( 36.27 )^{2} ≈ 53

hectictar
Nov 2, 2017

#1**+1 **

Best Answer

1) When y = 0 , x = 1, 3, or -4

A polynomial in that form with zeros at 1, 3, and -4 is

y = a(x - 1)(x - 3)(x + 4)

To find a , plug in -1 for x and 2 for y .

2 = a(-1 - 1)(-1 - 3)(-1 + 4)

2 = a(-2)(-4)(3)

2 = 24a

a = 1/12

So the polynomial is y = (1/12)(x - 1)(x - 3)(x + 4) And here's a graph to check it.

2) We want to know what p is when supply = demand . That is, when

6 + 0.001p^{3} = 80 - 0.02p^{2}

Let's solve this using a graph. The p value that makes the equations have the same y value is

p ≈ 36.27

Look at the graph or plug this value for p into either function to find out how many hair dryers are supplied and demanded at this price.

D( 36.27 ) = 80 - 0.02( 36.27 )^{2} ≈ 53

hectictar
Nov 2, 2017

#5**+1 **

.001p^3 + 6 = -.02p^2 + 80

p^3 + 6000 = -20p^2 + 80000

p^3 + 20p^2 - 74000 = 0

Note, AT.......there IS a "formula" for finding cubic root(s)...but...it's EXTREMELY messy....!!!!!

The "actual" real-number answer is :

p = 10/3 (-2 + (991 - 3 sqrt(109113))^(1/3) + (991 + 3 sqrt(109113))^(1/3))

Let's just go with hectictar's solution...!!!!!

CPhill
Nov 2, 2017

#6**+1 **

After seeing that long and obnoxious answer, I think my teacher will be okay with me using my graphing calculator.

Thanks CPhill!!!

AdamTaurus
Nov 2, 2017