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 α , β  And γ 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 = -Coefficient of x2coefficient of x3

 α + β  +  γ   = -36

 α + β  +  γ  = -12                                      -------------------  ( 1 )

Sum of the products of zeros taken two at a time = Coefficient of xcoefficient of x3
 α β  +  β ‚Äčγ  + γ α   = -56             ---------------------  ( 2 )

Product of zeros  =  -Constant termcoefficient of x3
 α β γ   = -16                                   --------------------- ( 3 )

Now for value of 

 α-1  + β-1  + γ-1

We can simplify it As :

1α+ 1β+1γ
Taking L.C.M. and get

αβ + βγ + γαα β γ

Now substitute values from equation 2 and 3 we get

-56-16

 α-1 + β-1 + γ-1    =  5                                           ( Ans )

 

  • 185

The given polynomial is

6x3 + 3x2 - 5x + 1

Here a = 6, b = 3, c = -5 and d = 1

α + β + γ = -b/a = -3/6 = -1/2

αβγ = -d/a = -1/6

α-1 + β-1 + γ-1 = 1/α + 1/β + 1/γ

= (α + β + γ)/ αβγ

= (-1/2)/(-1/6)

= 3

  • -22
What are you looking for?