help with zeros of a polynomial function?

Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
-3i and √3



My question:
I know that 3i is another zero, but is -√3 also a zero?

Answer 1

No, -sqrt(3) does not have to be a zero. Check this polynomial out:

(x+3i)(x-3i)(x-sqrt(3))

It has real coeffs and the two roots you want.

Answer 2

If you were looking for INTEGER or RATIONAL coefficients, -sqrt(3) would have to be a zero.

But since you're looking for REAL coefficients, it doesn't.

On the other hand, if a complex number is the root of a real polynomial, so is its conjugate. So any real polynomial that has 3i as a root will also have -3i as a root, and vice-versa.

So the answer is (x+3i)(x-3i)(x-sqrt(3)) = (x^2 + 9)(x - sqrt(3))

Depending on your teacher's preferences, you might or might not want to multiply that out for your final answer.