CBSE Class 11-science Answered
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 = - [ p3 - 4p3 + 16 pq - 64 r ] = 3 p3 - 16 p q + 64 r
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