Finding degree of nth polynomial

Question

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value.

n=3

2 and 5i are zeros;

f(-1)=156

 

f(x)=___________________

Answer 1

If the coefficients are real and one of the zeros is a complex number, then its complex conjugate must also be a zero.  For n = 3, the zeros are:

 

2

5i

-5i

 

The 3rd degree polynomial is

 

f(x) = A(x - 2)(x - 5i)(x + 5i), 

 

with A ≠ 0.

 

f(x) = A(x - 2)(x² + 25)

 

f(-1) = 156

 

156 = A(-1 - 2)[(-1)² + 25]

 

156 = -3A(26) = -78A

 

A = -156/78 = -2

 

f(x) = -2(x - 2)(x² + 25)

 

You can expand this if you wish.