Question

How do I use the conjugate zeros theorem?

Answer 1

As a student, if a teacher tells you that a polynomial with real coefficients has `3i` for one of its zeros, that you can reason:

With real corrifients, if `3i` is a zero, then so is `-3i`. So I know that `(x-3i)` and `(x-(-3i)) = (x+3i)` are both factors.

So I know that `(x-3i)(x+3i) = x^2 + 9` is a factor. That might help me factor the polynomial. (If I've learned division of polynomials.)

I saw this interesting bit of reasoning recently here on Socratic.

We know that a polynomial of degree `n` has `n` zeros (counting multiplicity). So a polynomial of odd degree has an odd number of zeros.
We also know that if a polynomial has real coefficients, the any imaginary zeros occur in conjugate pairs.

So we can conclude that a polynomial with real coefficients and odd degee must have at least one real zero. (An odd number of zeros cannot all be imaginary.)