Question

If $\alpha$ and $\beta$ are the roots of the equation $x^{2} + 2x + 4 = 0$, then $\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}$ is equal to

• A. $-1/2$
• B. $1/2$
• C. $32$
• D. $1/4$

Answer 1

The Correct Option is D

Given equation is $x^{2} + 2x +4 = 0$ Since $\alpha, \beta$ are roots of this equation. $\therefore\, \alpha+\beta=-2$ and $\alpha\beta=4$ Now, $\frac{1}{\alpha^{3}}+\frac{1}{\beta^{3}}$ $=\frac{\alpha^{3}+\beta^{3}}{\left(\alpha\beta\right)^{3}}$ $=\frac{\left(\alpha+\beta\right)\left(\alpha^{2}+\beta^{2}-\alpha\beta\right)}{\left(\alpha\beta\right)^{3}}$ $=\frac{\left(-2\right)\left(\left(\alpha+\beta\right)^{2}-3\alpha\beta\right)}{4\times4\times4}$ $=\frac{-2\left(4-12\right)}{4\times4\times4}$ $=\frac{\left(-2\right)\times\left(-8\right)}{4\times4\times4}=\frac{1}{4}$