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If alpha,beta,gamma are the roots of the...

If `alpha,beta,gamma` are the roots of the equation `x^3-p x+q=0,` ten find the cubic equation whose roots are `alpha//(1+alpha),beta//(1+beta),gamma//(1+gamma)` .

Text Solution

Verified by Experts

The correct Answer is:
`(p + q -1)-x^(3)- (2p +3q)x^(2) + (p+ 3q)x - q = 0`
Let ` y = (alpha )/(1 + alpha ) rArr alpha = (y)/(1 -y)`
since `alpha ` is a root of the equation ` x^(3) - px + q = 0`, so
` alpha ^(3) - palpha + q = 0`
`rArr (y^(3))/((1 - y)^(3)) - p (y)/(1 -y) + q =0`
or ` y^(3) - py (1 - y)^(2) + q (1 - y)^(3) = 0`
or `(1 - p - q )y^(3) + (2p + 3q)y^(2) - (p+ 3q)y + q = 0`
Therefore , the reuired cubic equation is
`(p + q - 1)x^(3) - (2 p + 3q)x^(2) + )(p+ 3 ) x - q = 0` .
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