Question

If $\alpha$ is a root of $x^4+x^2-1=0$, the value of $({\alpha }^6+2{\alpha }^4{)}^{2012}$ is?

• A. $0$
• B. $-1$
• C. $1$
• D. None of these

Answer 1

The correct option is C $1$

$x^4+x^2-1=0$

root= $\alpha$

$\Rightarrow {\alpha }^4+{\alpha }^2-1=0$

$\Rightarrow {\alpha }^4=1-{\alpha }^2$

$\Rightarrow ({\alpha }^6+2{\alpha }^4{)}^{2012}=({\alpha }^4({\alpha }^2+2){)}^{2012}$

$=\left(1\right(1-{\alpha }^2\left)\right(2+{\alpha }^2){)}^{2012}$

$=(2+{\alpha }^2-2{\alpha }^2-{\alpha }^4{)}^{2012}$

$=(-{\alpha }^4-{\alpha }^2+2{)}^{2012}$

$=(2-{\alpha }^2-1+{\alpha }^2{)}^{2012}=1$

$\therefore ({\alpha }^6+2{\alpha }^4{)}^{2012}=1$ .