Question

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

Answer 1

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