There exists $P\in \mathbb{R}[X]$ such that : $P(J)^n=I_p+J$?

Question

Let $J\in \mathcal{M}_{p}(\mathbb{R})$ defined by: $$ J=\left [ \begin{matrix} 0& 0& 0& \dots &0&0\\ 1& 0& 0&\dots &0&0\\ 0 & 1& 0& \dots &0&0 \\ \vdots & \vdots& \vdots& \vdots& \vdots\\ 0 & 0&0&\dots&1&0\\ \end{matrix} \right ] $$ Why, for $n\geq 1$, there exists $P\in \mathbb{R}[X]$ such that : $P(J)^n=I_p+J$? Any idea please.