Question

Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 8−i, and passing through the origin.

Answer 1

Passing through the origin means that 0 is a root
for a polynomial with real coefients (an integer is real), if a+bi is a root, a-bi is also a root

so zeroes are
i,-i 8-i, 8+i and 0

the factors of an equation with factors r1,r2,r3 is
(x-r1)(x-r2)(x-r3)

so
zeroes of i,-i,8-i, 8+i, 0 is
(x-i)(x+i)(x-(8-i))(x-(8+i))(x-0)
expanded
$x^5-16x^4+66x^3-16x^2+65x$

the polynomial is $x^5-16x^4+66x^3-16x^2+65x$