For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $t _ { n } + s _ { n } = 6 + n$ and $t _ { n } s _ { n } = 8 + n$.
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by
$$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$
We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set
$$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$
Show that $t _ { n } + s _ { n } = 6 + n$ and $t _ { n } s _ { n } = 8 + n$.