Question

if alpha and Beth's are the roots of the equation x^2+3x+2=0 then alpha^5+betta^5​

Answer 1

Given f(x)→ x²+3x+2= 0

To Find

• $\sf { \alpha }^{5} + { \beta }^{5}$

Solution

$\\\\{ \sf \: f(x) \implies x²+3x+2= 0} \ \\ \\ \ { \sf \implies x²+2x + x+2= 0} \\ \\ { \sf \: \: \:\implies x(x+2 )+1( x+2)= 0} \\ \\ { \sf \: \: \: \: \implies (x + 1)(x + 2)= 0~~~~~~~~~~~}$

$\\ Let \\ \: \alpha = - 1 \: \: \: and \: \: \: \beta = - 2 \: \: \: \: \: \: \: \: \: \: \:$

$\\ \sf{Now \: to \: find \: the \: value \: of \: { \alpha }^{5} + { \beta }^{5} }$

$\sf{\rightsquigarrow \alpha }^{5} + { \beta }^{5} = {( - 1)}^{5} + {( - 2)}^{5}$

${\rightsquigarrow \alpha }^{5} + { \beta }^{5} = { - 1 - 32 }$

${\rightsquigarrow\sf \alpha }^{5} + { \beta }^{5} = \bf{ - 33 }$