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Question
If α, β, γ and δ are the roots of the polynomial equation 2x4 + 5x3 – 7x2 + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + δ and αβγδ
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Solution
The given equation is 2x4 + 5x3 – 7x2 + 8 = 0.
÷ 2 ⇒ `x^4 + 5/2 x^3 - 7/2 x^2 + 4` = 0
Let the roots be α, β, γ, δ
α + β + γ + δ = ` - 5/2`
αβγδ = – 4
To form the quadratic equation with the given roots α + β + γ + δ, αβγδ.
x2 – x(S.O.R) + P.O.R = 0
`x^2 - x((-5)/2 - 4) + ((-5)/2) (- 4)` = 0
⇒ `x^2 - x((-13)/2) + 10` = 0
2x2 + 13x + 20 = 0
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