$P(x)=x^5+x^2+1$ has roots $x_1, x_2, x_3, x_4, x_5$ and $g(x)=x^2-2$ then $g\left(x_1\right) g\left

Question

$P(x)=x^5+x^2+1$ has roots $x_1, x_2, x_3, x_4, x_5$ and $g(x)=x^2-2$ then $g\left(x_1\right) g\left(x_2\right) g\left(x_3\right) g\left(x_4\right) g\left(x_5\right)-30 g\left(x_1 x_2 x_3 x_4 x_5\right)=$

Professor PC Thomas Classes · Answer

Let $y=x^{2}-2=g(x) \Rightarrow x=\sqrt{y+2}$. $P(\sqrt{y+2})=0 \Rightarrow y^{5}+20y^{4}+40y^{3}+79y^{2}+74y+23=0$. $g(x_{1})\dots g(x_{5})=-23$. $x_{1}\dots x_{5}=-1$.