Question

How do you find a fourth degree polynomial given roots `2+i` and `1-sqrt5`?

Answer 1

`x^4 -6x^3 +10x^2 +2x -15`

Explanation:

If 2+i is a root, its conjugate 2-i would also be a root. Likewise, if `1-sqrt5` is a root, its conjugate `1+sqrt5` would also be root. The polynomial would thus be

`(x -2-i)(x-2+i)(x-1+sqrt5)(x-1-sqrt5)`

`=((x-2)^2 +1) ((x-1)^2 -5)`

`=(x^2 -4x +5)(x^2 -2x -3)`

`=x^4 -6x^3 +10x^2 +2x -15`