Q.If alpha beta gamma are zeroes …

{tex}\alpha , \beta \text { and } \gamma{/tex} are zeroes of the polynomial 6x³ + 3x² - 5x + 1

in the given polynomial,  6x³ + 3x² - 5x + 1

a=6,  b=3,  c=-5,  d=1

Sum of the roots = {tex}- \frac {b}{a}{/tex}
{tex}\alpha + \beta + \gamma = - \frac { 3 } { 6 }{/tex}
{tex}\alpha + \beta + \gamma = - \frac { 1 } { 2 }{/tex}

sum of the Product of the roots = {tex}\frac {c}{a}{/tex}
{tex}\alpha \beta + \beta \gamma + \gamma \alpha = - \frac { 5 } { 6 }{/tex}

Product of the roots = {tex}- \frac{d}{a}{/tex}

 {tex}\alpha \beta \gamma = - \frac { 1 } { 6 }{/tex}
{tex}\therefore \quad \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } = \frac { \alpha \beta + \beta \gamma + \gamma \alpha } { \alpha \beta \gamma }{/tex}
{tex}= \frac { - 5 / 6 } { - 1 / 6 } = \frac { - 5 } { 6 } \times \frac { 6 } { - 1 }{/tex}
Hence, {tex}\alpha ^ { - 1 } + \beta ^ { - 1 } + \gamma ^ { -1 } = 5{/tex}