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if α, ß, Γ be the roots of the equation x^3+px^2+qx+r=0 then find the value of (α+ß-3Γ)(ß+Γ-3α)(α+Γ-3ß)
Asked by nitinkrjsr07082008 | 15 Apr, 2024, 11:32: PM

If α , β and γ are roots of equation , then

( x- α ) ( x - β ) ( x - γ ) = 0

x3 - ( α + β + γ ) x2 +  ( α β + β γ + γ α) x - ( α β γ ) = 0 ................(1)

given equation is

x3 + p x2 + q x + r = 0 ...........................(2)

By comparing eqn.(1) , (2) and (3) , we get

( α + β + γ ) = - p ..............................(3)

( α β + β γ + γ α) = q .......................(4)

( α β γ ) = - r .................................(5)

Let δ = ( α + β - 3 γ ) (  β + γ - 3 α ) (  γ + α  - 3β )

Using eqn.(3) , above equation is written as

δ = - ( p + 4 γ ) ( p + 4 α ) ( p + 4β )

δ = -[ p3 + 4 ( α + β + γ ) p2 + 16 ( α β + β γ + γ α)  p + 64 ( α β γ )  ]

By using eqn.(3) , (4) and (5) , we re write above expression as

δ = - [ p- 4p3 + 16 pq - 64 r ] = 3 p3 - 16 p q + 64 r

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Answered by Thiyagarajan K | 16 Apr, 2024, 04:02: PM

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