Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 3- i, and passing through the origin

Question

baru · Answer

consider an arbitrary degree 5 polynomial
$$Ax^5+Bx^4+Cx^3+Dx^2+Ex+F=0$$

baru · Answer

question says leading co-efficient =1
so A=1

baru · Answer

$$y=x^5+Bx^4+Cx^3+Dx^2+Ex+F$$
actually consider this

baru · Answer

(0,0) is a point on it.
substitute and get F=0

baru · Answer

$$y=x^5+Bx^4+Cx^3+Dx^2+Ex+0$$

baru · Answer

well now you have 4 unknowns,

and you are given 2 roots.

subsitute them, and eliminate 2 of the unknowns,

i guess there is no unique answer to this problem, you can choose what ever you want for the remaining variables

baru · Answer

sorry ignore that, i just read up something
conjugate root theorem: if a+bi is a root, then a-bi is also a root

baru · Answer

so you actually have roots : i, -i, 3-i, 3+i

four roots and four unknowns