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

Question

n=4

2i and 5i are the zeros

f(-1)= 130

find f(x)

Answer 1

The complex conjugates of the two complex zeros must also be zeros, so your four zeros are

2i

-2i

5i

-5i

and your polynomial is

f(x) = A(x - 2i)(x + 2i)(x - 5i)(x + 5i),

with A ≠ 0.

f(x) = A(x²+ 4)(x² + 25)

Now set

f(-1) = 130

130 = A[(-1)² + 4][(-1)² + 25]

Solve for A and finish from here.