Answer:
[tex]f(x)=x^4+15x^2-16[/tex]
Step-by-step explanation:
It is given that the leading coefficient of the polynomial is 1.
We need to find a polynomial function f of the least degree.
If c is a root of a polynomial then (x-c) is a factor of the polynomial.
The given zeros are -1, 1, -4i, 4i. So,
[tex]f(x)=(x-1)(x+1)(x+4i)(x-4i)[/tex]
[tex]f(x)=(x^2-1^2)(x^2-(4i)^2)[/tex]
[tex]f(x)=(x^2-1)(x^2+16)[/tex] [tex][\because i^2=-1][/tex]
[tex]f(x)=x^2(x^2+16)-1(x^2+16)[/tex]
[tex]f(x)=x^4+16x^2-x^2-16[/tex]
[tex]f(x)=x^4+15x^2-16[/tex]
Therefore, the required polynomial is [tex]f(x)=x^4+15x^2-16[/tex] .
