Let a , b , c be the the roots of the equation x 3 + x 2 − 333 x − 1002 = 0 . If P = a 3 + b 3 + c 3 then the value of 2006 P =

Question

Let a , b , c be the the roots of the equation x 3 + x 2 − 333 x − 1002 = 0 . If P = a 3 + b 3 + c 3 then the value of 2006 P ​ =

TestHub · Accepted Answer

Let α be the root of the given cubic where α can take values a , b , c Hence α 3 + α 2 − 333 α − 1002 = 0 or α 3 = 1002 + 333 α − α 2 ∴ Σ α 3 = Σ1002 + 333Σ α − Σ α 2 = 3006 + 333Σ α − [ ( Σ α ) 2 − 2Σ α 1 ​ α 2 ​ ] But Σ α = − 1 ; Σ α 1 ​ α 2 ​ = − 333 ∴ a 3 + b 3 + c 3 = 3006 − 333 − [ 1 + 666 ] = 3006 − 333 − 667 = 3006 − 1000 = 2006 = P

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