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Asked by bhaveshjshah11 | 05 Dec, 2023, 09:43: AM
If multiple number of questions are asked in one posting , only first question will be answered.

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f (x) = ( x- α ) ( x - β ) ( x - γ )= x3-  ( α + β + γ ) x2 + (αβ + βγ + γα ) x - (αβγ)  ...........................(1)

given function f(x) =  x3 - β x2 - (α/2) x - γ  .............................. (2)

By comparing coefficients of x2  of eqn.(1) and eqn.(2) , we get

( α + β + γ ) = β  ,  hence  we get ,  α + γ = 0 ...............(3)

By comparing coefficients of x  of eqn.(1) and eqn.(2) , we get

(αβ + βγ + γα ) = - α/2

β( α + γ ) + γ α = - α/2

Using eqn.(3) , we get from above expression as  γ α = - α/2   or  γ = -1/2

if γ = -1/2  , then from eqn.(3) , we get α = 1/2

By comparing constant terms  of eqn.(1) and eqn.(2) , we get

(αβγ) = γ  or  α β = 1

if α = 1/2 and   (α β)  = 1    ,  then β = 2  .

Then given polynomial is

f(x) = ( x - (1/2) ) ( x - 2 ) ( x + (1/2) )

f(x) = ( 3 - (1/2) )  ( 3 -2 ) ( 3 + (1/2)) = (5/2) (7/2) = 35/4

Answered by Thiyagarajan K | 05 Dec, 2023, 02:49: PM

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