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If the roots of equation x^(3) + ax^(2) ...

If the roots of equation `x^(3) + ax^(2) + b = 0 are alpha _(1), alpha_(2), and `
` alpha_(3) (a , b ne 0)`. Then find the equation whose roots are
`(alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(2)alpha_(3)+alpha_(3)alpha_(1))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(1)alpha_(3)+alpha_(1)alpha_(2))/(alpha_(1)alpha_(2)alpha_(3)) `.

Text Solution

Verified by Experts

The correct Answer is:
`bx^(3) + ax - 1 = 0 `
`(alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3))+ (-alpha_(1) alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)) = - (1)/(alpha_(2))`
Therefore, the required has roots `-1//alpha_(1),-1//alpha_(2),-1//alpha_(3)`.
`therefore y = (1)/(x) or x = - (1)/(y)`
Hence , the required equation is
`(-(1)/(y))^(3) + a (-(1)/(y))^(2) + b = 0`
`rArr by^(3) + ay - 1 = 0`
`rArr bx^(3) + ax - 1 = 0` .
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