Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0,$ ${\alpha }^9+{\beta }^9=$

Answer 1

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

$\Rightarrow \alpha +\beta =2$

$\alpha \cdot \beta =4$

Now, ${\alpha }^9+{\beta }^9=({\alpha }^3+{\beta }^3{)}^3-3{\alpha }^3{\beta }^3({\alpha }^3+{\beta }^3)$

$=\left[\right(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta ){]}^3-3\left(\alpha \beta {)}^3\right((\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta \left)\right)$

$=[2^3-3(4\left)\right(2){]}^3-3(4{)}^3\left(\right(2{)}^3-3\left(4\right)\left(2\right))$

$=(8-24{)}^3-192(8-24)$

$=-4096+3072$

$=1024$

$=2^9$