Consider the following theorem appearing in Linear Algebra Done Right
4.15 Polynomials with real coefficients have zeros in pairs.
Suppose $p\in P(\mathbb{C})$ is a polynomial with real coefficients. If $\lambda\in\mathbb{C}$ is a zero of $p$, then so is $\bar{\lambda})$.
The proof is quite simple. If we have a polynomial $p\in P(\mathbb{C})$ then it has form
$$\sum_{i=0}^m a_i z^i$$
with the $a_i\in\mathbb{R}$.
Suppose $\lambda$ is a root of $p$.
$$\sum_{i=0}^m a_i \lambda^i=0$$
And taking the complex conjugate of both sides
$$\sum_{i=0}^m a_i \bar{\lambda}^i=0$$
so $\bar{\lambda}$ is a root.
My question is about when $\lambda$ is a real number.
It seems that the only value of the theorem in that case is, for example, if we have a third degree polynomial and we find one complex root, then we know that the complex conjugate is the third root.
If we have a polynomial of degree $m$ and one real root, does the theorem give us any useful information?
It does not seem to tell us anything about the multiplicity of real roots, for example, correct?