n = 3; - 5 and i are zeros; f(-3) = 60

Question

Find an nth degree polynomial function with real coefficients satisfying the given conditions.

n = 3; - 5 and i are zeros; f(-3) = 60

Answer 1

For the coefficients to be real, the complex conjugate of a complex zero must also be a zero, so the zeros are

 

-5

i

-i

 

f(x) = A(x + 5)(x - i)(x + i) = A(x + 5)(x² + 1)

 

60 = A(-3 + 5)[(-3)² + 1] = A(2)(10) = 20A ⇒

 

A = 3

 

f(x) = 3(x + 5)(x² + 1)

 

You may expand to standard form if you wish.