Algebra polynomial function

Question

find a polynomial function of lowest degree with rational coefficients that has the numbers 3/4, -5 root squared and 3i amongst its zeros

Answer 1

The complex conjugate of a complex zero must also be a zero. In addition, square roots must appear in pairs of ±, so you have 5 zeros:

3/4

-√5

√5

3i

-3i

f(x) = A(x - 3/4)(x + √5)(x - √5)(x - 3i)(x + 3i)

f(x) = A(x - 3/4)(x² - 5)(x² + 9)

where A is real, rational, and A ≠ 0. You can expand to standard form if you wish.