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# Polynomial Functions

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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.

Nov 1, 2017
edited by AdamTaurus  Nov 1, 2017

#1
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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 + 0.001p3  =  80 - 0.02p2

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

Nov 2, 2017

#1
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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 + 0.001p3  =  80 - 0.02p2

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
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Is there any way you could solve #2 algebraically?

AdamTaurus  Nov 2, 2017
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I tried to but I couldn't get it.....someone else on here might know a way though.

hectictar  Nov 2, 2017
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Thanks anyway! CPhill is working on it.

AdamTaurus  Nov 2, 2017
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.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...!!!!!   Nov 2, 2017
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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
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Adam:  "Mathematica 11" does find a solution for p=\$36.27 and the number of hairdryers =~54, but as CPhill says it gets to be very"messy"!!.

Nov 2, 2017
edited by Guest  Nov 2, 2017