Question

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

n=4;
i and 3 i are zero;
f(-2)=65

f(x)=
An expression using x as the variable. Simplify your​ answer.

Answer 1

all irrational and complex zeros come in conjugal pairs,
so if i is a zero , so is -i
if 3i is a zero , so is -3i
so we have f(x) = a(x-i)(x+i)(x-3i)(x+3i)
f(x) = a(x^2 + 1)(x^2 + 9)
also f(-2) = 65
a(4+1)((4+9) = 65
65a = 65
a = 1

f(x) = (x^2 + 1)(x^2 + 9)
or
f(x) = x^4 + 10x^2 + 9

check:
http://www.wolframalpha.com/input/?i=solve+x%5E4+%2B+10x%5E2+%2B+9%3D0