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

Question

n=4;

i and 2i are zeros;

f(-1)=30

Answer 1

f(x) = a(x-i)^2(x-2i)^2 = a(x^2+1)(x^2+4)

x=i

x^2=-1

x^2+1=0

x=2i

x^2=-4

x^2+4 = 0

factors multiplied together are (x^2+1)(x^2+4)

substitute -1 for x, and solve for "a" the leading coefficient of the highest powered term

f(-1)=30 =a(1+1)(1+4) = a(2)(5)= 10a

a = 30/10 =3

f(x)= 3(x^2+1)(x^2+4) = 3(x^4 +5x^2 +4)

f(x) = 3x^4+15x^2+ 12