if alpha beta and gamma are the zeroes of polynomial 6x3+3x2-5x+1 then find the value of alpha raised to -1+beta raised to -1+gamma raised to -1

Answer:
We have $\alpha$ , $\beta$  And $\gamma$ are the zeros of cubic polynomial 6x³ + 3x² - 5x + 1
We know by the relationship between zeros and coefficient of a cubic polynomial , is
Sum of zeros = $-\frac{Coefficient of x^2}{coefficient of x^3}$

 $\alpha$ + $\beta$  +  $\gamma$   = $-\frac{3}{6}$

 $\alpha$ + $\beta$  +  $\gamma$  = $-\frac{1}{2}$                                      -------------------  ( 1 )

Sum of the products of zeros taken two at a time = $\frac{Coefficient of x}{coefficient of x^3}$
 $\alpha$ $\beta$  +  $\beta$ ​$\gamma$  + $\gamma$ $\alpha$   = $\frac{-5}{6}$             ---------------------  ( 2 )

Product of zeros  =  $-\frac{Cons\tan t term}{coefficient of x^3}$
 $\alpha$ $\beta$ $\gamma$   = $-\frac{1}{6}$                                   --------------------- ( 3 )

Now for value of 

 $\alpha$^-1  + $\beta$^-1  + $\gamma$^-1

We can simplify it As :

$\Rightarrow$$\frac{1}{\alpha }+ \frac{1}{\beta }+\frac{1}{\gamma }$
Taking L.C.M. and get

$\Rightarrow$$\frac{\alpha \beta + \beta \gamma + \gamma \alpha }{\alpha \beta \gamma }$

Now substitute values from equation 2 and 3 we get

$\Rightarrow$$\frac{{\displaystyle \frac{-5}{6}}}{-{\displaystyle \frac{1}{6}}}$

$\Rightarrow$  $\alpha$^-1 + $\beta$^-1 + $\gamma$^-1    =  5                                           ( Ans )