Question

If a polynomial function with rational coefficients has the zeros -1+i, sqrt5, what are the additional zeros?

Answer 1

The additional zeros are `(-1-i)` and `-sqrt5`.

Explanation:

Complex zeros always occur in pairs in the form of complex conjugates `a+-bi`.

The complex conjugate of `(-1+i)` is `(-1-i)`.

Similarly, zeros containing square roots must also come in pairs. If the polynomial has a zero of `sqrt5`, it must also have a zero of `-sqrt5`.

Think about the quadratic formula `x=frac{-b+-sqrt(b^2-4ac)}{21}`.

If the discriminant `b^2-4ac` is negative, there will be two complex zeros because of the `+-` signs before the square root. These are complex conjugates.

If the discriminant is positive but not a perfect square, there will also be two zeros of the form `+-sqrt`