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If α , β , γ are the roots of the equation x3 − 6 x2 + 11 x + 6 = 0, then Σ α2 β + Σ α β2 =
- a)80
- b)84
- c)90
- d)-84
Correct answer is option 'B'. Can you explain this answer?
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If α , β , γ are the roots of the equation x3 −...
Given Equation:
x^3 - 6x^2 + 11x + 6 = 0
Sum of Products of Roots taken two at a time:
Let α, β, γ be the roots of the given equation.
α + β + γ = 6 (from the coefficient of x^2 term)
αβ + βγ + γα = 11 (from the coefficient of x term)
αβγ = -6 (from the constant term)
Finding the Required Sum:
We need to find the value of Σα^2β + Σαβ^2.
Σα^2β + Σαβ^2 = (α^2 + β^2)γ + (β^2 + γ^2)α + (γ^2 + α^2)β
= (α + β)^2γ + (β + γ)^2α + (γ + α)^2β - 3αβγ
= (6 - γ)^2γ + (6 - α)^2α + (6 - β)^2β + 3αβγ
= 36(α + β + γ) - 12(αβ + βγ + γα) + α^3 + β^3 + γ^3
Using Sum of Cubes Formula:
α^3 + β^3 + γ^3 = 3αβγ + 6(α + β + γ)
= 3(-6) + 6(6)
= 12
Substitute the Values:
Σα^2β + Σαβ^2 = 36*6 - 12*11 + 12
= 216 - 132 + 12
= 84
Therefore, the required sum Σα^2β + Σαβ^2 = 84. Hence, the correct answer is option B.
x^3 - 6x^2 + 11x + 6 = 0
Sum of Products of Roots taken two at a time:
Let α, β, γ be the roots of the given equation.
α + β + γ = 6 (from the coefficient of x^2 term)
αβ + βγ + γα = 11 (from the coefficient of x term)
αβγ = -6 (from the constant term)
Finding the Required Sum:
We need to find the value of Σα^2β + Σαβ^2.
Σα^2β + Σαβ^2 = (α^2 + β^2)γ + (β^2 + γ^2)α + (γ^2 + α^2)β
= (α + β)^2γ + (β + γ)^2α + (γ + α)^2β - 3αβγ
= (6 - γ)^2γ + (6 - α)^2α + (6 - β)^2β + 3αβγ
= 36(α + β + γ) - 12(αβ + βγ + γα) + α^3 + β^3 + γ^3
Using Sum of Cubes Formula:
α^3 + β^3 + γ^3 = 3αβγ + 6(α + β + γ)
= 3(-6) + 6(6)
= 12
Substitute the Values:
Σα^2β + Σαβ^2 = 36*6 - 12*11 + 12
= 216 - 132 + 12
= 84
Therefore, the required sum Σα^2β + Σαβ^2 = 84. Hence, the correct answer is option B.
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If α , β , γ are the roots of the equation x3 − 6 x2 + 11 x + 6 = 0, then Σ α2β + Σ α β2=a)80b)84c)90d)-84Correct answer is option 'B'. Can you explain this answer?
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