Answer
Please see the proof below.
Work Step by Step
For a polynomial with real coefficients and odd degree $2n+1$, we can factor it into $2n+1$ linear terms containing real and complex zeros.
Based on the Conjugate Zeros Theorem, complex zeros occur in pairs, so the sum of the above linear terms with
complex zeros must be even. As the total degree is odd, it means that there must be at least one term containing
real zeros.
In other words, we proved that a polynomial with real coefficients and odd degree has at least one real zero.
