Question

If $\alpha ,\beta$ are the roots of $x^2-2x+4=0$, then ${\alpha }^5+{\beta }^5=$

• A. $8$
• B. $16$
• C. $32$
• D. $64$

Answer 1

The correct option is C $32$

$\alpha ,\beta$ are the roots of $x^2-2x+4=0$
$\Rightarrow \alpha +\beta =2,\alpha \beta =4$
${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
$=2^2-2\left(4\right)$
$=4-8=-4$
${\alpha }^3+{\beta }^3=(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )$
$=2^3-3\left(4\right)\left(2\right)$
$=8-24=-16$
${\alpha }^5+{\beta }^5=({\alpha }^3+{\beta }^3)({\alpha }^2+{\beta }^2)-{\alpha }^3{\beta }^2-{\alpha }^2{\beta }^3$
$=({\alpha }^3+{\beta }^3)({\alpha }^2+{\beta }^2)-\left(\alpha \beta {)}^2\right(\alpha +\beta )$
$=(-16)(-4)-\left(4{)}^2\right(2)$
$=64-32=32$