I have a question related to Lemma 4.2 from this pdf (which is, btw quite a nice exposition of Hoffman Singleton work on the classifications of Moore graphs of diameter 2 and 3.)
We are given a $n \times n$ real symmetric matrix $A$ with eigenvalues $d,\lambda_1,\lambda_2$ where $d$ has multiplicity 1. The key part of Lemma 4.2 computes the multiplicity of $\lambda_1$ and $\lambda_2$ when these two values are irrational.
The part I don't understand is when it tries to deduce that if $\lambda_1$ and $\lambda_2$ are irrational then they have both equal multiplicity $\frac{n-1}{2}.$ Somehow I don't understand the part when it says that if $(x-\lambda_1)$ divides $p(x)$ then $(x-\lambda_1)(x-\lambda_2)$ also divisides $p(x).$
How does that follows? Is anyone able to clarify the proof of the Lemma to me?