Question

Let $\alpha ,\beta ,\gamma$ are roots of the equation $x^3+qx+q=0$ then find the value of $(\alpha +\beta {)}^{-1}(\beta +\gamma {)}^{-1}+(\gamma +\alpha {)}^{-1}$ ?

• A. $1$
• B. $-1$
• C. $0$
• D. none

Answer 1

Answer: (A)

$1$

Let $\alpha ,\beta ,\gamma$ are roots of the quadratic equation $x^3+qx+q=0$

$\because a=1,b=0,c=qandd=q$

By relation between roots and coefficient of cubic equations we get,

$\alpha +\beta +\gamma =\frac{-0}{1}=0\alpha \beta +\beta \gamma +\gamma \alpha =\frac{q}{1}=q\alpha \beta \gamma =\frac{-q}{1}=-qA/Q,{(\alpha +\beta )}^{-1}+{(\beta +\gamma )}^{-1}+{(\gamma +\alpha )}^{-1}=\frac{1}{(\alpha +\beta )}+\frac{1}{(\beta +\gamma )}+\frac{1}{(\gamma +\alpha )}=\frac{{(\alpha +\beta +\gamma )}^2+(\alpha \beta +\beta \gamma +\gamma \alpha )}{(\alpha +\beta +\gamma )(\alpha \beta +\beta \gamma +\gamma \alpha )-\alpha \beta \gamma }=\frac{0+q}{0.q-(-q)}=1$