Constructing a polynomial function

Question

Construct a polynomial functionf(x) that has all the following properties (there is more than one possible answer):

f has degree 6,

f has four real zeros (not all distinct),

2 is a zero of multiplicity 3,

f behaves like y= 2x^6for large values of |x|,

and f(0) = 8.

Answer 1

One possibility is:

 

  f(x)  = C(x-2)(x-2)(x-2)(x-1)(x^2+1)  where C is a fixed number constant

 

  f(0) = 8 means that

 

 

   C(-2)(-2)(-2)(-1)(1) = 8

 

   8C = 8

 

  C = 8/8 = 1

 

f(x) = (x-2)(x-2)(x-2)(x-1)(x^2+1)