Question

Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros. 3-i, square root of 7

Answer 1

Real coefients
if a+bi is a root then a-bi is also a root
since 3-i is a root then 3+i s also a root

also the √7, so if √7 is a root then -√7 is also a root
so

f roots are r1 and r2 then the factored form is
f(x)=(x-r1)(x-r2)

roots are 3-i, 3+i and √7 and -√7

f(x)=(x-(3-i))(x-(3+i))(x-√7)(x-(-√7))
expanded
f(x)=x⁴-6x³+3x²+42x-70