First slide

Theory of equations

Question

If $α, β, γ$ are the roots of the equation $x^{3} + 4 x + 1 = 0$ , then ${(α + β)}^{- 1} + {(β + γ)}^{- 1} + {(γ + α)}^{- 1} =$

Moderate

A

2
B

3
C

4
D

5

Solution

If $α, β, γ$ are the roots of the equation.
$x^{3} - {px}^{2} + qx - r = 0$
$α + β + γ = 0 ∴ {(α + β)}^{- 1} + {(β + γ)}^{- 1} + {(γ + α)}^{- 1} = \frac{1}{- α} + \frac{1}{- β} + \frac{1}{- γ} = - \frac{α β + β γ + γ α}{α β γ} = - \frac{4}{- 1} = 4$

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