<s>[INST] Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$ Let $\alpha$, $\beta$, $\gamma$, $\delta$ be the roots of $$z^4-2z^3+z^2+z-7=0$$ then find value of $$(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$$
Are Vieta's formulas appropriate?
 [/INST] transform the equation so that you get the desired roots.
so the given equation has roosts $\alpha,  \ \beta,\ \gamma, \ \delta $
 first transform $x \to \sqrt{x}$ so the new equation ha roots  $\alpha^2,  \ \beta^2,\ \gamma^2, \ \delta^2 $ then transform $x \to x-1$ to get the equation whose roots are $\alpha^2+1,  \ \beta^2+1,\ \gamma^2+1, \ \delta^2+1 $
now the answer is product of roots of new equation.
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