need more explanations on the proof about zeros of orthogonal polynomials

Question

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I do not understand why should not the polynomial $P_n(x)(x-x_1)...(x-x_n)$ change sign ?

Answer 1

Let $x_1,\ldots,x_p$ the roots of $p_n$ with even multiplicity and $x_{p+1},\ldots,x_n$ the roots of $p_n$ with odd multplicity, we can write : $$ p_n(x)=\lambda\prod_{i=1}^p(x-x_i)^{2m_i}\prod_{i=p+1}^n(x-x_i)^{2m_i+1} $$ with $\lambda\in\mathbb{R}$ and $m_i\in\mathbb{N}$ for all $i$. Thus $$ p_n(x)(x-x_{p+1})\ldots(x-x_n)=\lambda\underbrace{\prod_{i=1}^p(x-x_i)^{2m_i}\prod_{i=p+1}^n(x-x_i)^{2(m_i+1)}}_{\geqslant 0} $$