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Question
If α, β and γ are the roots of the cubic equation x3 + 2x2 + 3x + 4 = 0, form a cubic equation whose roots are 2α, 2β, 2γ
Solution
Given that α, β, γ are the roots of x3 + 2x2 + 3x + 4 = 0
Compare with x3 + bx2 + cx + d = 0
b = 2, c = 3, d = 4
α + β + γ = – 6 = – 2
αβ + βγ + γα = c = 3
αβγ = – d = – 4
Given roots are 2α, 2β, 2γ
2α + 2β + 2γ = 2(α + β + γ)
= 2(– 2)
= – 4
(2α)(2β) + (2β)(2γ) + (2γ)(2α) = (4αβ + 4βγ + 4γα)
= 4(αβ + βγ + γα)
= 4(3)
= 12
(2α)(2β)(2γ) = 8(αβγ)
= 8(– 4)
= – 32
The equation is
x3 – x2 (2α + 2β + 2γ) + x(4αβ + 4βγ + 4γα) – 8(αβγ)
= 0
⇒ x3 – x2 (– 4) + x(12) – (– 32) = 0
⇒ x3 + 4x2 + 12x + 32 = 0
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