Question: 3+i, 3 Lowest Degree. Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.
When a polynomial has complex roots, they always come as a pair.
So if x = 3+i is one root, then x = 3 - i is another root.
(Two complex roots will always be of the form a + ib, a - ib. )
So we have at least three roots: x = 3, x = 3 + i, x = 3 - i.
Creating the polynomial from these three roots.
(x - 3)(x - (3 + i))(x - (3 - i)) = 0
(x - 3)(x^2 - 3x + 3i - 3x - 3i + 9 + 1) = 0
(x - 3)(x^2 - 6x + 10) = 0
x^3 - 6x^2 + 10x - 3x^2 + 18x - 30 = 0
x^3 - 9x^2 + 28x - 30 = 0