Construct a polynomial function with the following properties: fifth degree, 2 is a zero of multiplicity 2, -5 is the only other zero, leading coefficient is 4

Question

Construct a polynomial function with the following properties: fifth degree, 2 is a zero of multiplicity 2, −5 is the only other zero, leading coefficient is 4

Answer 1

f(x) = 4(x+5)³(x-2)²

 

  = 4(x³+15x²+75x+125)(x²-4x+4)

 

  = 4(x⁵-4x⁴+4x³+15x⁴-60x³+60x²+75x³-300x²+300x+125x²-500x+500)

 

  = 4(x⁵+11x⁴+19x³-115x²-200x+500)

 

  = 4x⁵+44x⁴+76x³-460x²-800x+2000

Answer 2

A polynomial of degree 5 has a leading term of Cx⁵, with C being a coefficient.  The number of zeros must be at most 5.  Multiplicity means the number of times a zero appears.  Odd multiplicity means the zero crosses the x-axis.  Even multiplicity means the zero touches the x-axis, but never crosses it.

 

 

Here are the zeros with their multiplicities.

 

x = 2 , multiplicity of 2        

x = -5 , multiplicity of 1   

 

 

Okay, lets start to plan out our function by writing it in factored form.

 

f(x) = 4(x³ + 125)(x - 2)(x - 2)

 

f(x) = (4x³ + 500)(x² - 4x + 4)

 

f(x) = 4x³(x² - 4x + 4) + 500(x² - 4x + 4)

 

f(x) = 4x⁵ - 16x⁴ + 16x³ + 500x² - 2000x + 2000