Question

Let $x_1,x_2,x_3,x_4$ and $x_5$ be roots of $p\left(x\right)=x^5+x^2+1$ and $g\left(x\right)=x^2-2$. The value of $g\left(x_1\right)g\left(x_2\right)g\left(x_3\right)g\left(x_4\right)g\left(x_5\right)-30g\left(x_1{x}_2{x}_3{x}_4{x}_5\right)$ is___

Answer 1

$x_1{x}_2{x}_3{x}_4{x}_5=-1\Rightarrow g\left(x_1{x}_2{x}_4{x}_5\right)=1-2=-1$
Let $y=g\left(x\right)\Rightarrow x=\sqrt{y+2}$
Substitute in $p\left(x\right)\Rightarrow (y+2{)}^{\frac{5}{2}}=-(y+3)$
$\Rightarrow y^5+{10}^4+40y^3+79y^2+74y+23=0$
Roots are:$g\left(x_i\right),p=1$ to 5
$\therefore g\left(x_1\right)g\left(x_2\right)g\left(x_3\right)g\left(x_4\right)g\left(x_5\right)=-23\therefore -23-30(-1)=7$