Finding the nth term to a polynomial

Question

n=4

i and 2i are zeros

f(-2)=120

someone please explain the process on how to solve this as i am very confused, thank you :)

Answer 1

i and 2i are zeros, imaginary zeros come in conjugate pairs, so -i and -2i are also zeros

x=i, x^2 = -1 x^2 +1 is a factor

x= 2i, x^2 = -4 x^2+4 is another factor multiply the 2 factors to get

(x^2 +1)(x^2 +4) = x^4 +5x^2 + 4

add another unknown constant factor a to get a(x^2+1)(x^2+4)

f(-2) = 120 plug in -2 for x and set the 3 factor product = 120

a(5)(8) = 120, a = 120/40 = 3

the 4 degree polynomial is: 3(x^4 + 5x^2 +4) = 3x^4 +15x^2 +12

if you really meant find the nth term with n=4, there is no 4th term, just 3 terms. Above is how problems like this are typically solved