Question

α, β, γ are zeroes of cubic polynomial x3 – 2x2 + qx – r. If α + β = 0 then show that 2q = r.

Answer 1

α+β+r=2

=r=2

αβ+βr+αr=q

αβ+r(0)=q

αβ=q

αβr=r

2q=r

Answer 2

Answer:

To prove  --> 2q=r

Step-by-step explanation:

α +β+γ=2 ............................ eqn 1

αβ+βγ+γα=q........................... eqn 2

αβγ=r ....................................... eqn 3

Since:

α+β=0

Therefore from eqn 1 --> γ=2

Also: substituting γ in eqn 3

αβ*2=r

αβ=r/2............. eqn 4

Now LHS:

2q=2 (αβ+βγ+γα)........................ (from eqn 2)

    = 2(αβ+2β+2α)................................ putting the value of γ.

    = 2( r/2+2β+2α)

    =  r + 4β+4α

    =r + 4(α+β)

    = r + 4(0)

    = r = RHS

Hence Proved.

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