Find the nth degree

Question

N=4

i and 3 i are zeros;

f(-2)=65

Answer 1

If i is a root so is -i

That means the associated factors are (x - i) and (x + i)

If 3i is a root, so is -3i

That means the associated factors are (x - 3i) and (x + 3i)

Multiply all 4 of these together. But there can be a multiplier (A) in front of the polynomial:

f(x) = A (x - i) (x + i) (x - 3i) (x + 3i)

Plug in x = -2 and f(x) = 65 to find A

That will give you the entire equation

Answer 2

+or- i are zeros

+or- 2i are zeros

the polynomial eats (-2,65)

(x+i)(x-i)(x-3i)(x+3i) =

(x^2 + 1)( x^2 + 9) =

(x^4 + 10x^2 + 9)

P(x) = k(x^4 + 10x^2 + 9)

65 = k ( (-2)^4 + 10(-2)^2 + 9)

65 = k ( 16 + 40 + 9)

65 = k ( 65)

k = 1

since k=1, it just multiplies out

P(x) = (x^4 + 10x^2 + 9)