8. Show that there exists a unique family $q _ { 0 } , \ldots , q _ { n - j - 1 }$ of monic polynomials of $\mathbb { R } [ \boldsymbol { X } ]$ such that $\operatorname { deg } \left( q _ { i } \right) = i$ for $0 \leqslant i \leqslant n - j - 1$ and such that for all $0 \leqslant i \neq i ^ { \prime } \leqslant n - j - 1$,

$$\left\langle q _ { i } , q _ { i ^ { \prime } } \right\rangle _ { j } = 0 .$$

8t. We set $q _ { n - j } = \prod _ { i = j + 1 } ^ { n } \left( X - r _ { i } \right)$. Show that $q _ { n - j }$ is the unique monic polynomial of degree $n - j$ satisfying, for all $0 \leqslant i \leqslant n - j - 1$,

$$\left\langle q _ { i } , q _ { n - j } \right\rangle _ { j } = 0$$

Let $2 \leqslant i \leqslant n - j$. Show that there exist real numbers $a _ { i }$ and $b _ { i }$ such that

$$q _ { i } - X q _ { i - 1 } = a _ { i } q _ { i - 1 } + b _ { i } q _ { i - 2 }$$

9b. Show that

9c. Show that $b _ { i } < 0$.

$$b _ { i } \left\langle q _ { i - 2 } , q _ { i - 2 } \right\rangle _ { j } = - \langle \underbrace { X q _ { i - 1 } , q _ { i - 2 } } _ { \geqslant 0 } \rangle _ { j } j .$$

\begin{enumerate}
  \setcounter{enumi}{101}
  \item For $i \in \{ 0,1 \}$, show that the polynomial $q _ { i }$ has exactly $i$ roots in $\mathbb { R }$ (note that we do not require the roots to belong to the interval $I$ ).
\end{enumerate}

10b. Show that, for all $1 \leqslant i \leqslant n - j$, the polynomial $q _ { i }$ has exactly $i$ distinct real roots, that these roots are simple and that if $x _ { 1 } < x _ { 2 }$ are two consecutive roots of $q _ { i }$, there exists a unique root of $q _ { i - 1 }$ in the interval $] x _ { 1 } , x _ { 2 } [$.\\
10c. Deduce that, for all $0 \leqslant i \leqslant n - j - 1$, we have $q _ { i } \left( r _ { j + 1 } \right) > 0$.\\
For $0 \leqslant i \leqslant n - j - 1$, there therefore exists a unique real number $\alpha _ { i }$ such that

$$q _ { i + 1 } \left( r _ { j + 1 } \right) + \alpha _ { i } q _ { i } \left( r _ { j + 1 } \right) = 0$$

We fix $0 \leqslant i \leqslant n - j - 1$ and we set

$$p _ { i } = \frac { q _ { i + 1 } + \alpha _ { i } q _ { i } } { X - r _ { j + 1 } }$$

We denote $c _ { 0 } , \ldots , c _ { i } \in \mathbb { R }$ the coordinates of $p _ { i }$ in the basis $\left( q _ { 0 } , \ldots , q _ { i } \right)$ of $\mathbb { R } _ { i } [ X ]$.\\
11a. Show that, for $0 \leqslant \ell \leqslant i$,

$$\left\langle q _ { i + 1 } + \alpha _ { i } q _ { i } , \frac { q _ { \ell } - q _ { \ell } \left( r _ { j + 1 } \right) } { X - r _ { j + 1 } } \right\rangle _ { j } = 0 .$$

11b. Show that, for every integer $0 \leqslant \ell \leqslant i$, there exists a real $\gamma _ { \ell } > 0$ such that $c _ { \ell } = \gamma _ { \ell } c _ { 0 }$ and deduce that $c _ { \ell } > 0$.\\