Let $p_n$ be a real polynomial of exactly degree $n$. Suppose that $p_n$ has atleast $n-1$ distinct real roots of odd multiplicity. It is claimed that $p_n$ has $n$ simple real roots. How do you verify this claim?
Here is what I thought: if those roots were of multiplicity $\geq 3$, then $p_n$ will have atleast $n+1$ roots counting multiplicities (I hope I do math correctly). Also, there can't be a non-real root, in which case that conjugated one would also be a root, and so in total $n+1$ roots. Something like that.