# How do I use the conjugate zeros theorem?

Apr 26, 2015

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

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

So I know that $\left(x - 3 i\right) \left(x + 3 i\right) = {x}^{2} + 9$ 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.)