Question

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of $x^4+x^2+1=0$, then the equation whose roots are ${\alpha }^2,{\beta }^2,{\gamma }^2,{\delta }^2$, is:

• A. $(x^2-x+1{)}^2=0$
• B. $(x^2+x+1{)}^2=0$
• C. $(x^4-x^2+1{)}^2=0$
• D. $x^2-x+1=0$

Answer 1

The correct option is B $(x^2+x+1{)}^2=0$

As $x^4+x^2+1=0\Rightarrow x^2=\frac{-1\pm i\sqrt{3}}{2}$
Then equation with roots $\left(\frac{-1+i\sqrt{3}}{2}\right),\left(\frac{-1-i\sqrt{3}}{2}\right)$ is
$y^2+y+1=0$
Hence ${\alpha }^2,{\beta }^2,{\gamma }^2,{\delta }^2$ are roots of
$(x^2+x+1{)}^2=0$