A) \[c\]
B) \[-c\]
C) \[-3c\]
D) \[3c\]
Correct Answer: D
Solution :
Let \[\alpha ,\beta ,\gamma \] are the roots of the equation \[{{x}^{3}}+bx+c=0\] Then, \[\Sigma \alpha =0,\,\,\Sigma \alpha \beta =b\] and \[\alpha \beta \gamma =-c\] ?.(i) Now, \[\Sigma \alpha \,\,\Sigma \alpha \beta =(\alpha +\beta +\gamma ).(\alpha \beta +\beta \gamma +\gamma \beta )\] \[=\Sigma {{\alpha }^{2}}\beta +3\alpha \beta \gamma \] \[\Rightarrow \] \[\Sigma {{\alpha }^{2}}\beta =\Sigma \alpha \Sigma \alpha B-3\alpha \beta \gamma =0.\Sigma \alpha \beta -3(-c)\] [from Eq. (i)] \[=3c\]You need to login to perform this action.
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