Question

Suppose $Î\pm ,β,γ$ are the roots of
$x^3+x^2+x+2=0.$ Then, the value of
$\left(\frac{Î\pm +β−2γ}{γ}\right)\left(\frac{β+γ−2Î\pm }{Î\pm }\right)\left(\frac{γ+Î\pm −2β}{β}\right)$ is

• A. $−\frac{47}{2}$
• B. $\frac{47}{2}$
• C. $−47$
• D. 47

Answer 1

Answer: (A)

Since, $Î\pm ,β,γ$ are the roots of $x^3+x^2+x+2=0$
$∴Î\pm +β+γ=−1$
Now, $\frac{Î\pm +β−2γ}{γ}=\frac{−1−γ−2γ}{γ}=\frac{−1−3γ}{γ}$
$∴x=\frac{−1−3γ}{γ}⇒γ=\frac{−1}{x+3}$
Now, the equation whose root is $\frac{−1}{x+3}$, is
${\left(\frac{−1}{x+3}\right)}^3+{\left(\frac{−1}{x+3}\right)}^2+\left(\frac{−1}{x+3}\right)+2=0$
$⇒−1+(x+3)−(x+3{)}^2+2(x+3{)}^3=0$
$⇒−1+x+3−x^2−9−6x+2x^3+54+18x^2+54x=0$
$⇒2x^3+17x^2+49x+47=0$
$∴\left(\frac{Î\pm +β−2γ}{γ}\right)\left(\frac{β+γ−2Î\pm }{Î\pm }\right)\left(\frac{γ+Î\pm −2β}{β}\right)=\frac{−47}{2}$