Find the condition that `x^3-px^2+qx-r=0` may have sum of its two roots is zero .

Explanation

Let `alpha,beta ,gamma` be the roots of
`x^3-px^2+qx-r=0`
`alpha+beta +gamma=p " "…..(1) `
`alphabeta+betagamma+gammaalpha=q" "……(2)`
`alphabetagamma=r" "………(3) `
Given `alpha +beta=0 `
(`because ` sum of two roots is zero )
From (1) , `gamma=p`
`because gamma ` is a root of `x^3-px^2 +qx-r=0 `
`gamma^3-pgamma^2+qgamma-r=0`
But `gamma=p`
`rArr p^3-p(p^2)+q(p)-r=0`
`rArr p^3-p^3+qp-r=0 `
`therefore`qp=r is the required condition .