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

Question

n=4; i and 3i are zeros; f(-2)=130

Answer 1

To obtain real coefficients, the complex conjugate of any complex zero must also be a zero. That makes for the following 4 zeros:

i

-i

3i

-3i

The form of the polynomial is

f(x) = A(x - i)(x + i)(x - 3i)(x + 3i) = A (x² + 1)(x² + 9)

f(-2) = 130 ⇒

130 = A[(-2)² + 1][(-2)² + 9] = A(5)(13) = 65A

A = 130/65 = 2

f(x) = 2(x² + 1)(x² + 9)

You can expand this to standard form if you wish.