given the polynomial f=4x^3-12x^2+mx+n find m, n if f(x)=0 for x=i

Given the polynomial f=4x^3-12x^2+mx+n find m, n if f(x)=0
for x=i.

If f(i) = 0, then i is the root of
4x^3-12x^2+mx+n.

Since complex roots appear in pairs , if i
is a root then -i os also a root..

Therefore f(x) the
factors, (x+i)(x-i) = x^2+1.

Therefore  we can write
4x^3-12x^2+mx+n =
(x^2+1)(ax+b).

4x^3-12x^2+mx+n = ax^3+bx^2+ax+b.

We
equate the coefficients of like terms:

a = 4,  b=
=-12,

m = a = 4

n = b =
-12.

Therefore  m = 4 and n =
-12.

Therefore
4x^3-12x^2+mx+n = x^3-12x^2+4x-12.