If α,β,γ are non zero roots of x3+px2+qx+r=0, then the equation whose roots are α(β+γ),β(γ+α),γ(α+β)
A
x3−2qx2+(pr+q2)x+(r2−pqr)=0
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B
x3−2qx2+(pr+q2)x+(r2+pqr)=0
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C
x3+2qx2+(pr+q2)x+(r2−pqr)=0
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D
x3+2qx2+(pr+q2)x+(r2+pqr)=0
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Solution
The correct option is Ax3−2qx2+(pr+q2)x+(r2−pqr)=0 Given equation: x3+px2+qx+r=0
We know that, α+β+γ=−pαβ+βγ+αγ=qαβγ=−rr≠0(∵α,β,γ≠0)
Let y=α(β+γ)=q−αβγα=q−(−r)α ⇒y−q=rα⇒α=ry−q
which is a root of given equation.
Therefore, the equation whose roots are α(β+γ),β(γ+α),γ(α+β) is,