Question

If roots of $x^3+5x^2-7x-1=0$ are $\alpha ,\beta ,\gamma$, then the equation whose roots are $\alpha \beta ,\beta \gamma ,\gamma \alpha$, is

• A. $x^3-7x^2+5x+1=0$
• B. $x^3+7x^2-5x-1=0$
• C. $x^3+5x^2+7x+1=0$
• D. $noneofthese$

Answer 1

Answer: (B)

$x^3+7x^2-5x-1=0$

$x^3+5x^2-7x-1=0$

$\Rightarrow$ $\alpha \beta \gamma =1$

The equation whose roots are

$\frac{1}{\alpha },\frac{1}{\beta },\frac{1}{\gamma }$ is $\frac{1}{x^3}+\frac{5}{x^2}-\frac{7}{x}-1=0$

$-x^3-7x^2+5x+1=0$

$x^3+7x^2-5x-1=0$

$\alpha \beta ,\beta \gamma ,\gamma \alpha$

We can write

$\frac{\alpha \beta }{\alpha \beta \gamma }.\frac{\beta \gamma }{\alpha \beta \gamma }.\frac{\gamma \alpha }{\alpha \beta \gamma }$ ....($\because \alpha \beta \gamma =1$)

$\frac{1}{\gamma },\frac{1}{\alpha },\frac{1}{\beta }$

So, equation whose roots are

$\alpha \beta ,\beta \gamma ,\gamma \alpha \Rightarrow \frac{1}{\gamma },\frac{1}{\alpha },\frac{1}{\beta }$ is

$x^3+7x^2-5x-1=0$