Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 5 - i, and passing through the origin.

For a polynomial with real cofieints, if a+bi is a root, a-bi is also a root

zeros, i and 5-i

passes through origin means 0 is also a zero

get plus and minus of the roots

i, - 1, 5-i, 5+i and 0 are roots

for a poly with roots, r1, r2, r3, r4, r5, the facotred form is

(x-r1) (x-r2) (x-r3) (x-r4) (x-r5)

sub the roots

(x-i) (x - (-i)) (x - (5-i)) (x - (5+i)) (x-0) =

(x-i) (x+i) (x-5+i)) (x-5-i)) (x) =

x (x^2+1) (x^2-10x+26) =

x^5-10x^4+27x^3-10x^2+26x

the polynomial is f (x) = x^5-10x^4+27x^3-10x^2+26x