Find the cubic polynomial with zeros 3 and 3+4i ( and 3-4i) and having f(1) = 64

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 ;

3 and 3+4i are zeroes

f(1)=64

Answer 1

Since 3+4i is a root, 3-4i must also be a root.

 

f(x) = a(x-3)[x-(3+4i)][x-(3-4i)]

 

      = a(x-3)[(x-3)-4i][(x-3)+4i]

 

      = a(x-3)[(x-3)²-(4i)²]

 

      = a(x-3)(x²-6x+25)

 

Since f(1) = 64, -40a = 64.  So, a = -8/5

 

f(x) = (-8/5)(x-3)(x²-6x+25)