Question

please guys help ...............

Answer 1

$3 {x}^{2} + 5x + 7 \\ \alpha + \beta = \frac{ - 5}{3} \\ \alpha \beta = \frac{7}{3} \\ { \alpha }^{3} + { \beta }^{3} = ( \alpha + \beta )( { \alpha }^{2} + { \beta }^{2} - \alpha \beta ) \\ = ( \alpha + \beta )( { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta - 2 \alpha \beta - \alpha \beta ) \\ = (\alpha + \beta)(( { \alpha + \beta )}^{2} - 3 \alpha \beta ) \\ = ( \frac{ - 5}{3} )( ({ \frac{ - 5}{3} )}^{2} - 3( \frac{7}{3} )) \\ = \frac{ - 5}{3} ( \frac{25}{9} - 7) \\ = \frac{ - 5}{3} ( \frac{25 - 63}{9} ) \\ = \frac{ - 5}{3} ( \frac{ - 38}{9} ) \\ = \frac{190}{27}$
$\frac{1}{ { \alpha }^{2} } + \frac{1}{ { \beta }^{2} } = \frac{ { \alpha }^{2} + { \beta }^{2} }{ { \alpha }^{2} { \beta }^{2} } \\ = \frac{ { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta - 2 \alpha \beta }{ { \alpha }^{2} { \beta }^{2} } \\ = \frac{( { \alpha + \beta )}^{2} - 2 \alpha \beta }{ { \alpha }^{2} { \beta }^{2} } \\ = \frac{ \frac{25}{9} - \frac{14}{3} }{ \frac{49}{9} } \\ = \frac{ \frac{25 - 42}{9} }{ \frac{49}{9} } \\ = \frac{ - 17}{49}$
$\frac{ \alpha }{ \beta } + \frac{ \beta }{ \alpha } + 5 \alpha \beta \\ = \frac{ { \alpha }^{2} + { \beta }^{2} + 5 { \alpha }^{2} { \beta }^{2} }{ \alpha \beta } \\ = \frac{( { \alpha + \beta )}^{2} - 2 \alpha \beta + 5 {( \alpha \beta )}^{2} }{ \alpha \beta } \\ = \frac{ \frac{25}{9} - \frac{14}{3} + \frac{245}{9} }{ \frac{7}{3} } \\ = \frac{ \frac{25 - 42 + 245}{9} }{ \frac{7}{3} } \\ = \frac{228 \times 3}{9 \times 7} \\ = \frac{228}{27} = \frac{76}{9}$