Question

if $\alpha ,\beta$ are roots of $x^2+2x+4=0$, then ${\alpha }^6+{\beta }^6=$

• A. $128$
• B. $64$
• C. $192$
• D. $-128$

Answer 1

The correct option is A $128$

$\alpha ,\beta$ are roots of $x^2+2x+4=0$

$⟹\Delta$ is negative

$\therefore$ roots are complex.

$\therefore \alpha =2\omega ,\beta =2{\omega }^2.$ where $w^3=1$

$\therefore {\alpha }^6+{\beta }^6$

$=2^6.{\omega }^6+2^6.{\omega }^{12}$

$=2^6({\omega }^6+{\omega }^{12})=2^6({\omega }^3.{\omega }^3+{\omega }^3{\omega }^3{\omega }^3{\omega }^3)=2^6(1+1)=2^6\times 2=2^7=128$

Answer- Option A