find a monic polynomial of lowest possible degree with rational coefficeients and having both 2 and 3-i roots.
If 3 - i is a root....so is 3 + i
So the polynomial is of the form
x^3 + bx^2 + cx + d
b = - sum of the roots = = -[ 2 + (3 + i) + (3 - i)] = - 8
c = sum of the product of roots taken two at a time = 2(3 + i) + 2 (3 -i) + ( 3 + i) (3 -i) = 6 + 6 + 9 - i^2 = 12 + 9 + 1 = 22
d = - product of the roots = - 2 (3+i) ( 3 - i) = - 2 ( 9 -i*2) = - 2 ( 9 + 1 ) = -20
The polynomial is
x^3 - 8x^2 + 22x - 20